Find the volume of the solid W in the octant x ≥ 0, y ≥ 0, z ≥ 0 bounded by x + y + z = 8 and x + y + 32 = 8.
Answers
The volume of the solid W in the first octant bounded by x + y + z = 8 is 256/3 cubic units.
To find the volume of the solid W in the octant x ≥ 0, y ≥ 0, z ≥ 0, we need to analyze the given equations:
1. x + y + z = 8
2. x + y + 32 = 8
First, simplify the second equation: x + y + 32 - 32 = 8 - 32, resulting in x + y = -24. This equation implies a negative value for x or y, which contradicts the conditions of the given octant (x ≥ 0, y ≥ 0, z ≥ 0). Therefore, we can disregard the second equation as it doesn't contribute to the boundaries of the solid W.
Now, we can focus on the first equation, x + y + z = 8, which represents a plane in the first octant. To find the volume of the solid W, we need to calculate the volume of the tetrahedron bounded by this plane and the coordinate axes.
The intersections of the plane with the coordinate axes are:
x + 0 + 0 = 8 → x = 8 (x-intercept)
0 + y + 0 = 8 → y = 8 (y-intercept)
0 + 0 + z = 8 → z = 8 (z-intercept)
These intercepts form a tetrahedron with vertices at (8,0,0), (0,8,0), (0,0,8), and (0,0,0). The volume of the tetrahedron can be calculated using the formula:
Volume = (1/3) * Base Area * Height
In this case, the base area is a right triangle with legs of length 8, so:
Base Area = (1/2) * 8 * 8 = 32
The height of the tetrahedron is equal to the z-intercept, which is 8:
Volume = (1/3) * 32 * 8 = 256/3
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Related Questions
factor the following
the equation is f(x) = x^3+27
Answers
The factored form of the cubic equation x³ + 27 is equal to (x + 3) · (x - 3 / 2 - i 3√3 / 2) · (x - 3 / 2 + i 3√3 / 2). The roots of the cubic equation are - 3, 3 / 2 + i 3√3 / 2 and 3 / 2 - i 3√3 / 2.
How to factor a cubic equation
In this problem we find a cubic equation of the form x³ + a³, whose factor form can be found by using algebra properties:
f(x) = x³ + 27
f(x) = (x + 3) · (x² - 3 · x + 9)
f(x) = (x + 3) · (x - 3 / 2 - i 3√3 / 2) · (x - 3 / 2 + i 3√3 / 2)
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Which point is the incenter of Triangle JKL ?
Enter your answer in the box.
point
Answers
Answer:
Point N. I took the test!
Step-by-step explanation:
It is found that in-center of the triangle JKL is point N.
What is a triangle?A triangle is a polygon that has three sides and three vertices. It is one of the basic figures in geometry.
We are given a triangle JKL.
In triangle JKL, they have drew perpendicular bisectors of each side of the triangle.
Also we are given the angle bisectors of the each of angles.
Since perpendicular bisector is a line that intersect a segment into two equal parts and also perpendicular to it.
Also we know that the in-center is the point forming the origin of a circle inscribed inside the triangle. which is constructed by taking the intersection of the angle bisectors of the three vertices of the triangle.
We can see that angle bisectors are intersecting at a point N.
Therefore, It is found that in-center of the triangle JKL is point N.
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round the decimal two hundred fourteen and five hundred fifty seven thousandths to the nearest tenth
Answers
Answer:
214.6
Step-by-step explanation:
214.557 -> 214.6
Two hundred fourteen and five hundred fifty-seven thousandths to the nearest tenth will be 214.6.
What is Algebra?Algebra is the study of graphic formulas, while logic is the interpretation among those signs.
Two hundred fourteen and five hundred fifty-seven.
The expression in mathematical form thousandths to the nearest tenth, we have
⇒ 514.557
⇒ 214.6
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help meeeeeee ASAP A triangle has side lengths of (r−10s) centimeters,(3r+2t) centimeters, and 2t−6s) centimeters. Which expression represents the perimeter, in centimeters, of the triangle?
−8s+4r−4t
−14st+6rt
4r+4t−16s
5rt−9rs−4st
Answers
The expression that represents the perimeter in centimetre of the triangle is 4r - 16s + 4t
Perimeter of a triangle
Perimeter of a triangle is the sum of the whole sides. The triangle has three sides. Therefore, the perimeter is the sum of the three sides.
The side length are as follows;
r - 10s3r + 2t2t - 6sTherefore,
perimeter = r - 10s + 3r + 2t + 2t - 6s
combine like terms
perimeter = r + 3r - 10s - 6s + 2t + 2t
perimeter = 4r - 16s + 4t
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Answer: If you use this equatin you can g3et to the number of 8343 this isvvery truie and you shoould mark it on your paperthank you mysos n
Step-by-step explanation:
please what should i pick
Answers
Answer:
A, B, D, E
Step-by-step explanation:
They all are negative numbers when solved except for C
Ellen has a pencil 2x + 1/5cm long. Tom has a pencil 5x + 2/5cm long. How much longer is Tom's pencil than Ellen's?
Answers
The difference of length of Tom's pencil from Ellen's pencil is given by the equation A = 3x + 1/5 cm
What is an Equation?Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given data ,
Let the difference of length be represented as A
Now , the equation will be
The length of Ellen's pencil is p = 2x + 1/5 cm
The length of Tom's pencil is q = 5x + 2/5 cm
Now , the difference of length A = q - p
Substituting the values in the equation , we get
The difference of length A = 5x + 2/5 cm - 2x + 1/5 cm
On simplifying the equation , we get
The difference of length A = ( 5x - 2x ) + ( 2/5 - 1/5 ) cm
The difference of length A = 3x + 1/5 cm
Therefore , Tom's pencil is longer than Ellen's pencil by 3x + 1/5 cm
Hence , the equation is A = 3x + 1/5 cm
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I don’t know if this is correct !!!!!!!!!! Please answer correctly !!!!!!!!!!! Will mark Brianliest !!!!!!!!!!!!!!!!!!!
Answers
Answer:
The first one is correct. But why can't they be proven congruent
Step-by-step explanation:
There were 450 students enrolled in a middle school at the beginning of the school year. The number of students enrolled in the school at the end of the year was 10 percent greater than the number of students enrolled in the school at the beginning of the year. How many students were enrolled in the school at the end of the year?
Answers
Answer:
450*1.10 is 495
Step-by-step explanation:
450+45=495
Or
450*1.10=495
a manager receives 9 applications for a specific position. she wants to narrow it down to 4. in how many ways can she rank 4 applications?
Answers
The number of ways that can she rank 9 applications should be 126.
What is the definition of combination in math?
A combination is a mathematical technique that determines the number of possible arrangements in a collection of items where the order of the selection does not matter. In combinations, you can select the items in any order.
Calculation of the number of ways:
Since A manager receives 9 applications for a specific position. She wants to narrow it down to 4.
So here we do apply the permutation here:
9*8*7*6/ 4*3*2
=126
Hence, The number of ways that can she rank 4 applications should be 126
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1.
Use the functions f(x) = 3x - 4 and g(x) = x2 + 2 to find these values.
a.
f(7)
b.
g(5)
C.
f(-5)
d.
9(-3)
e.
Find x when f(x) = 7
f.
Find x when g(x) = 18
Answers
Answers:
A. 17
B. 27
C. -19
D. -7
E. 17
F. 326
BRAINLIEST PLEASE
Help pls thank you :)
Answers
2. The hypotenuse is 13
b. Sin C = 0.5, Cos C = 0.875, TanC = 0.571
What is a right triangle?Finding the length of a side given the lengths of the other two sides, determining if a triangle is a right triangle, and other issues involving right triangles can be solved with the Pythagorean theorem.
The ratios of the side lengths in right triangles give rise to trigonometric functions such as sine, cosine, and tangent, which are used extensively in trigonometry and geometry.
Using;
\(c^2 = a^2 + b^2\\c = \sqrt{} 5^2 + 12^2\)
= 13
Sin C = 4/8
= 0.5
Cos C = 7/8
= 0.875
Tan C = 4/7
= 0.571
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1. Find the solution to the system of equations. 2x+3y=1
3x+ay=b
where a,b are real numbers. a) (5 pts) Give conditions on a and b such that the system is consistent. b) (5 pts) Give conditions on a and b such that the system has a unique solution. In this case, give the solution in terms of a and b. c) (5 pts) Give conditions on a and b such that the system has infinitely many solutions. Geometrically describe the solutions in this case.
Answers
a) The system of equations is consistent when its determinant is non-zero. Since the determinant of the coefficient matrix is -7, the system is consistent for all real numbers a and b.
b) The system of equations has a unique solution if and only if the determinant of the coefficient matrix is non-zero. Thus, the system has a unique solution for all real numbers a and b, except when a = 21/7 and b = 3, which would make the determinant equal to zero. If a = 21/7 and b = 3, then the system has infinitely many solutions.
c) The system of equations has infinitely many solutions if and only if the determinant of the coefficient matrix is zero and the system is consistent. If a = 21/7 and b = 3, then the system has infinitely many solutions. The solution set is a line with the equation y = (-2/3)x + 1/3. If the determinant is not zero, then the system is consistent and has a unique solution.
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Answer this question and show me how to check it
Answers
In order to rewrite these values in the standard form, let's calculate the product with the power of 10 from each number.
For the length, we have:
\(\begin{gathered} 8\cdot10^4 \\ =8\cdot10000 \\ =80000\text{ meters} \end{gathered}\)For the thickness, we have:
\(\begin{gathered} 5\cdot10^{-6} \\ =5\cdot0.000001 \\ =0.000005\text{ meters} \end{gathered}\)A cab charges customers a one time fee plus a fee per mile driven. If the cab drove someone for 7 miles and charged them $42 and also drove someone else for 10 miles and charged them $57. What is the rate of change in this scenario?
Answers
Answer:
7
Step-by-step explanation:
Justin bought 3 CDs that were each the same price. Including sales tax, he paid a total of $49.50 Each CD had a tax of $1.10 What was the price of each CD before tax?
Answers
Answer:
We know he bought 3 CDs and he paid 49.50 dollars, which the tax included. Now, we can calculate how much did he pay for each CD by dividing the total cost (49.50) by the number of CDs (3):
49.50/3 = 16.50
Now, we know each CD costs 16.50 with tax included so, if it has a tax of1.10 dollars, we only need to substract the price (16.50) and the tax (1.10)
16.50 - 1.10 = 15.40
Without tax, ech CD costs 15.4 dollars ($15.4)
Answer:
The price before sale tax is $15.40
Step-by-step explanation:
$1.10 × 3 = $3.30
$49.50 - $3.30 = $46.20
$46.20 ÷ 3 = $15.40
Which relation is a function?{(1, 1), (-1, 2), (0, 3), (1, 4), (-2, 5)}{(-1, -1), (-1, -2), (-1, -3), (-1, -4), (-1, -5)}{(-3, -1), (-2, 0), (-1, 1), (0, 2), (1, 3)}{(-3, -3), (-2, -2), (-1, -1), (0, 0), (-1, 1)}
Answers
Answer:
The 3rd relation is a function!
Answer:
(0, 3)
Step-by-step explanation:
As the Only function out of all
Evaluate ∂y
∂u
at (x,y,z)=(2,2,0) for the function u(p,q,r)=e pq
cosr;p= x
1
,q=x 2
lny,r=z u(p,q,r)=e pq
cosr;p= x
1
,q=x 2
lny,r=z olduğuna göre ∂y
∂u
nin(x,y,z)=(2,2,0) A. - −1 B. - 0 C. - 4 D. - 1 E. - 8
Answers
The value of ∂y/∂u at \((x,y,z) = (2,2,0) is 2/e^2.\)
Thus, the answer is D. - 1.
Given function is \(u(p,q,r)=e^pq cost;p= x1,q=x2lny,r=z\)
So, we can evaluate the partial derivative of y with respect to u at \((x, y, z) = (2, 2, 0)\) for the function u(p, q, r) as follows;∂
\(y/∂u = (∂u/∂y) / (∂y/∂u)\)
Therefore, we need to find the partial derivative of u with respect to y and the partial derivative of y with respect to u to solve the given problem.
So, let us begin with calculating the partial derivative of u with respect to y:
\(∂u/∂y=e^(pq) cosr(∂pq/∂y)+e^(pq) (-sinr) r(∂r/∂y)\\=e^(x2lny) cos(0) [(∂/∂y)x(2lny)]+e^(x2lny) (-sin0) (0)\\=e^(x2ln2) [2/x2]+0\\=e^2 [2/2^2]\\=e^2/2\)
On differentiating with respect to y, the given function u (p, q, r) is simplified as follows;
\(u(p,q,r)=e^(pq)cost;p= x1,q=x2lny,r=z\)
On differentiating the above equation with respect to y, we get
\(∂u/∂y=e^(pq) cosr(∂pq/∂y)+e^(pq) (-sinr) r(∂r/∂y)\\=e^(x^2lny)cos(0) [(∂/∂y)x(2lny)]+e^(x^2lny) (-sin0) (0)\\=e^(x^2ln2) [2/x^2]+0\\=e^2 [2/2^2]\\=e^2/2\)
Now, we can calculate the partial derivative of y with respect to u using the formula;
\(∂y/∂u = (1 / (e^(x^2ln2)/2)) \\= 2/e^2\)
The value of ∂y/∂u at\((x,y,z) = (2,2,0) is 2/e^2.\)
Thus, the answer is D. - 1.
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What is meant by zero-point energy?
Answers
Zero-point energy is the lowest energy state of a quantum system at absolute zero temperature.
Zero-point energy is the lowest energy state of a quantum system at absolute zero temperature. To calculate the zero-point energy, the energy of the system must first be determined at the lowest temperature that can be achieved in practice, typically close to absolute zero. The energy at this temperature is then subtracted from the energy of the system at higher temperatures. This difference is the zero-point energy. The zero-point energy can be calculated by taking the sum of the energies of all of the quantum states at absolute zero. This sum can be approximated using the uncertainty principle and the Heisenberg Uncertainty Principle. The sum is then multiplied by Planck's constant to get a value for the zero-point energy. Finally, the total zero-point energy is the sum of the individual zero-point energy values of each quantum state. This energy represents the energy of the system at absolute zero and is the lowest energy state possible.
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a brick is in the shape of a rectangular prism with a length of 8 inches, a width of 3.5 inches, and a height of 2 inches. the brick has a density of 2.7 grams per cubic centimeter. find the mass of the brick to the nearest gram a brick is in the shape of a rectangular prism with a length of 8 inches, a width of 3.5 inches, and a height of 2 inches. the brick has a density of 2.7 grams per cubic centimeter. find the mass of the brick to the nearest gram
Answers
The mass of brick is 2478 gram.
What is a rectangular prism?A rectangular prism has a total of 6 faces, 12 sides, and eight vertices. Like a cuboid, it has three dimensions- the base width, the height, and the length. The top and base of the rectangular prism are rectangular in shape. The pairs of opposite faces of a rectangular prism are identical or congruent.
A brick is in the shape of a rectangular prism with a length of 8 inches, a width of 3.5 inches, and a height of 2 inches.
The volume of rectangular prism = length×width×height
= 8×3.5×2
= 56
Thus volume of brick is 56 cubic inches.
Convert inches to centimeter
1 inch = 2.54 centimeter
Therefore,
56 cubic inches = 56 x 2.54 x 2.54 x 2.54 cubic centimeter
56 cubic inches = 917.676 cubic centimeter
Thus, we get,
volume of a rectangular prism = 917.676 cubic centimeter
The brick has a density of 2.7 grams per cubic centimeter
Density = 2.7 grams
The mass of brick is given by formula = density × volume
\(\Rightarrow\) = 2.7 × 917.676
\(\Rightarrow\) = 2477.72
\(\Rightarrow\) ≅ 2478
Thus mass of brick is 2478 gram.
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in the section, "Stride Rate Faster Than Usain Bolt's," what
does the author mean by "That is more than 10 times the
stride rate of Olympic Runner Usain Bolt? Use two details
from the article to support your response.
Answers
The author is comparing this stride rate to that of Olympic Runner Usain Bolt, who has a reported stride rate of 3.5 strides per second.
What is stride rate?The number of steps a runner takes in a specific length of time is referred to as stride rate and is commonly expressed in steps per second. You can figure it out by dividing the total number of steps you took during a certain amount of time by how long that time period lasted. Stride rate, which is frequently used as a gauge of running effectiveness, is impacted by a variety of variables, including running speed, terrain, and personal running form.
Hence, in the given section, the author is comparing this stride rate to that of Olympic Runner Usain Bolt, who has a reported stride rate of 3.5 strides per second.
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Lara bought 3 yards of fabric and a spool of thread for $28.10. If the spool of thread cost $1.25, how much did the fabric cost per yard? Write and solve an equation to answer.
Answers
Answer: $8.95 per yard of fabric
Step-by-step explanation:
Define the following:
x = cost of fabric per yard
Make an equation:
⇒ 3x + 1.25 = 28.10
⇒ 3x = 26.85
⇒ x = 8.95
∴ $8.95 per yard of fabric
I need help with this ( but I need to send another picture that has the answers )
Answers
Step 1:
Draw the figure
The diagonals of a rhombus bisect each other,
therefore
FJ = JH
this implies that
JH = 4.
Opposite angles of a rhombus are equal, so
The diagonals of a rhombus bisect the interior angles.
Therefore,
\(\text{ angle JHG = }\frac{120^0}{2}=60^0\)The diagonals of a rhombus are perpendicular
\(\begin{gathered} Hence, \\ \cos 60^0=\frac{4}{GH} \\ \Rightarrow GH=\frac{4}{\cos60^0}=\frac{4}{\frac{1}{2}}=4\times2=8 \end{gathered}\)GH = 8 (b)
Also,
Using the Pythagorean rule,
\(\begin{gathered} JG^2+4^2=GH^2 \\ \Rightarrow JG^2+16=8^2 \\ \Rightarrow JG=\sqrt[]{64-16}=\sqrt[]{48}=4\sqrt[]{3}=6.928\text{ } \\ \text{right choice is C} \end{gathered}\)FH = FJ + JH
this implies that
FH = 4 + 4 = 8
FH = 8
right choice B
The diagonals of a rhombus bisect the interior angles.
therefore
\(\begin{gathered} angle\text{ JFG = }\frac{angle\text{ IFG}}{2}=\frac{120^0}{2}=60^0 \\ \text{ right choice is h} \end{gathered}\)Sum of the interior angles of a rhombus is 360degrees.
And opposite interior angles of a polygon are congruent.
Therefore
\(2(\text{angle FGH) +2(angle IFG) = 360degr}ees\)\(\begin{gathered} \text{this implies that} \\ 2(\text{angle FGH) = 360 - 2(angle IFG)} \\ \Rightarrow\text{angle FGH = 180 -angle IFG = 180 - 120= }60^0 \\ \text{ right choice is h} \end{gathered}\)\(\begin{gathered} \text{ but }FGJ\text{ = }\frac{FGH}{2}\text{ (diagonals bisect interior angles)} \\ \Rightarrow\text{FGJ =}\frac{60^0}{2}=30^0 \\ \text{right choice is j} \end{gathered}\)Also,
\(\begin{gathered} \(\begin{gathered} \text{JGH}=\text{ }\frac{FGH}{2}=\frac{60}{2}=30^0 \\ right\text{ choice is j} \end{gathered}\)
What is the value of x?
(2x + 15)°
to
2x
A. 24
B. 33
C. 72
D. 75
Answers
Answer:
D. 75
Step-by-step explanation:
Which of the following is a right Riemann sum for arctan(1 + xdx? k=1 © ( aretan (1 + 4) :) į (aretan (4+4) ) ©Ë (arctan ( 1 + **) :) © (aretan (2 + %). :) arctan 1+ .
Answers
The right Riemann sum for arctan(1 + xdx) is Σ[arctan(1 + iΔx)]Δx, where i ranges from 1 to n and Δx is the width of each subinterval. The correct answer among the options provided is (arctan(1 + Δx) + arctan(1 + 2Δx) + ... + arctan(1 + nΔx))Δx.
In a right Riemann sum, the function is evaluated at the right endpoint of each subinterval. Therefore, we add up the values of arctan(1 + iΔx) at the endpoints of the subintervals, where i ranges from 1 to n. The width of each subinterval is Δx, so we multiply the sum by Δx to get the approximate value of the integral.
The provided expression (arctan(1 + Δx) + arctan(1 + 2Δx) + ... + arctan(1 + nΔx))Δx satisfies the conditions of a right Riemann sum, where the function is evaluated at the right endpoint of each subinterval. Therefore, this is the correct option among the given choices.
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What is the distance from L to M?
A. 10 units
B. 14 units
C. 100 units
D. 2 units
Answers
Explanation:
If there are outliers in a sample, which of the following is always true?a. Mean > Median
b. Standard deviation is smaller than expected (smaller than if there were no outliers)
c. Mean < Median
d. Standard deviation is larger than expected (larger than if there were no outliers)
Answers
In the presence of outliers in a sample, the statement that is always true is d. Standard deviation is larger than expected (larger than if there were no outliers).
Outliers are extreme values that are significantly different from the other data points in a sample. These extreme values have a greater impact on the standard deviation compared to the mean or median. As a result, the standard deviation increases when outliers are present. Therefore, option d is the correct answer.
To summarize, when outliers are present in a sample, the standard deviation is typically larger than expected, while the relationship between the mean and median can vary and is not always predictable.
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HELP ILL GIVE BRAINLIEST Write (2 + 3i)^3 in simplest a + bi form.
Answers
Answer:
18
Step-by-step explanation:
(2+3)^3 = (6)^3 = 18
Step-by-step explanation:
18
(2 + 3)^3=(6)^3=18
The CEO has left the company, so the stock prices have changed. The price has changed from $120 per share to $90 per share.
Answers
Answer:
120-90
=30
the change in stock price is $30 per share
a spinner has five equal sections labeled a, b, c, d, and e. a fair coin has faces labeled heads and tails. carlos will spin the arrow of the spinner and flip the coin one time each. what is the probability the arrow will land on the section labeled a and the coin will land on heads?
Answers
The probability of the arrow landing on section a is 1/5 since there are 5 equal sections. The probability of the coin landing on heads is 1/2 since there are only 2 possible outcomes (heads or tails) for the coin.
To find the probability of both events happening, we need to multiply the probability of the arrow landing on section a by the probability of the coin landing on heads. This gives us (1/5) * (1/2) = 1/10. So the probability that the arrow will land on the section labeled a and the coin will land on heads is 1/10. In other words, there is a 1 in 10 chance of both events happening. It is important to note that each event is independent of each other, meaning the outcome of one does not affect the other. This is because the spinner and the coin are not connected or related in any way.
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x - 1 = ( 2 - x ) ^2
Answers
Answer:
x=1
Step-by-step explanation:
x-1=(2-x)^2
x-1=4-2x-2x+x^2
x-1=4-4x+x^2
5x+x^2=5
x^2=1
x=1
Answer:
x = 2 , x = 3Step-by-step explanation:
x - 1 = ( 2 - x )²
Expand the terms in the bracket
That's
x - 1 = x² - 4x + 4
x² - 4x - x + 4 + 1 = 0
x² - 5x + 6 = 0
Solve the quadratic equation
x² - 5x + 6 = 0
(x - 2) (x - 3) = 0
x = 2 , x = 3Hope this helps you