Question 1

Solve the equation using factoring techniques. 3 w 3 $-$ 21 w 2 + 30 w = 0

Accepted Answer

First, factor out the greatest common factor, which is $3w$, giving $3w(w^2 - 7w + 10) = 0$. Then, factor the quadratic equation to get $3w(w-5)(w-2) = 0$. Setting each factor equal to zero gives the solutions $w = 0$, $w = 5$, and $w = 2$.

Question 2

The area of a triangular sheet of paper is 116 square inches. One side of the triangle is 5 inches more than three times the length of the altitude to that side. Find the length of that side and the altitude to the side.

Accepted Answer

Let the altitude be x and the length of the side be y. Then we have the equation "xy = 116" (since the area is 116 square inches). We also know that "y = 3x + 5" (since one side is 5 more than three times the altitude). Substituting y in terms of x into the first equation, we get:

x(3x + 5) = 116
3x^2 + 5x - 116 = 0

Solving for x using the quadratic formula, we get x = 8 or x = -4/3 (we ignore the negative solution since we're dealing with lengths). Plugging x=8 into y=3x+5, we get y=29. Therefore, the altitude is 8 inches and the length of the side is 29 inches.

Question 3

Solve the equation using factoring techniques. w 2 $-$ 3 w = 10 Please enter your answer as two numbers, separated by a comma.

Accepted Answer

The answer of Solve the equation using factoring techniques. w...

Question 4

Solve the equation using factoring techniques. x ² + 8 x + 7 = 0 Please enter your answer as two numbers, separated by a comma.

Accepted Answer

The answer of Solve the equation using factoring techniques. x...

Question 5

Factor the polynomial completely. $9 a ^ { 2 } - 24 a + 16$

Accepted Answer

We can start by trying to factor this polynomial by grouping. This involves finding two numbers that multiply to the constant term (16) and add to the linear coefficient (-24). These numbers are -2 and -8, so we can rewrite the polynomial as:
$$9a^2 - 24a + 16 = 9a^2 - 2a - 22a + 16$$
Then we can factor by grouping like this:
$$\begin{aligned} 9a^2 - 2a - 22a + 16 &= (9a^2 - 2a) + (-22a + 16) \ &= a(9a - 2) - 8(9a - 2) \ &= (a - 8)(9a - 2) \end{aligned}$$
So the polynomial is factored completely as $(a - 8)(9a - 2)$, which matches choice D.

Question 6

Solve the equation using factoring techniques. 2 n ³ = 8 n Please enter your answer as three numbers, separated by commas.

Accepted Answer

The answer of Solve the equation using factoring techniques. 2...

Question 7

Factor the polynomial completely. $9 x ^ { 2 } + 16$

Accepted Answer

The polynomial $9x^2 + 16$ is a sum of squares, which is not factorable over the real numbers.

Question 8

Solve the equation using factoring techniques. 3 t ( t $-$ 7 ) = 0 Please enter your answer as two numbers, separated by a comma.

Accepted Answer

The answer of Solve the equation using factoring techniques. 3...

Question 9

Factor the polynomial completely. $3 x + 3 y + a x + a y$

Accepted Answer

Factor out the common factor of 3 and a, which gives $3x+3y+ax+ay=3(x+y)+a(x+y)=(3+a)(x+y)$. Therefore, the correct choice is B, which shows the factors as $(x+y)(3+a)$.

Question 10

Solve the equation using factoring techniques. ( x + 3 ) 2 $-$ 25 = 0 Please enter your answer as two numbers, separated by a comma.

Accepted Answer

The answer of Solve the equation using factoring techniques. (...

Question 11

Factor the polynomial completely. $x ^ { 3 } - 8$

Accepted Answer

The polynomial $x^3 - 8$ is a difference of cubes, which can be factored as $(a^3 - b^3) = (a - b)(a^2 + ab + b^2)$. Here, $a = x$ and $b = 2$, so the factored form is $(x - 2)(x^2 + 2x + 4)$.

Question 12

Solve the equation using factoring techniques. 5 w 3 $-$ 60 w 2 + 175 w = 0 Please enter your answer as three numbers, separated by commas.

Accepted Answer

The answer of Solve the equation using factoring techniques. 5...

Question 13

Factor the polynomial completely. $2 n ^ { 3 } + 10 n ^ { 2 } - 16 n$

Accepted Answer

The answer of Factor the polynomial completely. \(2 n ^...

Question 14

The combined area of a square and a rectangle is 116 square centimeters. The width of the rectangle is 2 centimeters more than the length of a side of the square, and the length of the rectangle is 2 centimeters more than its width. Find the dimensions of the square and the rectangle.

Accepted Answer

Let x be the length of a side of the square. Then the width of the rectangle is x + 2 and the length of the rectangle is (x + 2) + 2 = x + 4. The combined area of the square and rectangle is x^2 + (x + 2)(x + 4) = 116. Simplifying this equation gives x^2 + 6x - 104 = 0, which can be factored as (x + 13)(x - 8) = 0. Since the length of a side of a square cannot be negative, x = 8 is the only solution. Therefore, the dimensions of the square are 8cm by 8cm, and the dimensions of the rectangle are 8cm + 2cm by 8cm + 4cm, or 10cm by 12cm. Thus, the answer is B.

Question 15

Factor the polynomial completely. $( x + 9 ) ^ { 2 } - y ^ { 2 }$

Accepted Answer

This is a difference of squares, so we can write it as $(x+9+y)(x+9-y)$.

Question 16

Factor the polynomial completely. $2 n ^ { 3 } + 6 n ^ { 2 } + 8 n$

Accepted Answer

Take out the greatest common factor of $2n$, which leaves $n^{2}+3n+4$. This trinomial cannot be factored further with real coefficients, so the final answer is $2n(n^{2}+3n+4)$.

Question 17

The perimeter of a rectangle is 32 inches and the area is 63 square inches. Find the length and width of the rectangle. Please enter your answer as two numbers, separated by a comma, without the units.

Accepted Answer

The answer of The perimeter of a rectangle is 32...

Question 18

Factor the polynomial completely. $1 - 36 n ^ { 2 }$

Accepted Answer

We can use the difference of squares formula: $$a^2 - b^2 = (a+b)(a-b)$$ In this case, let $a=1$ and $b=6n$. Then we have: $$1-36n^2 = (1)^2 - (6n)^2 = (1+6n)(1-6n)$$ So the polynomial is completely factored as $(1+6n)(1-6n)$, which corresponds to choice letter A.

Question 19

The Ortegas have an apple orchard that contains 70 trees. The number of trees in each row is 4 more than twice the number of rows. Find the number of rows and the number of trees per row.

Accepted Answer

Let x be the number of rows and y be the number of trees per row. Then we have the equations:

xy = 70  (the total number of trees)
y = 2x + 4  (the number of trees per row is 4 more than twice the number of rows)

Substituting the second equation into the first equation, we get:

x(2x + 4) = 70
2x^2 + 4x - 70 = 0
x^2 + 2x - 35 = 0
(x + 7)(x - 5) = 0
x = -7 or x = 5

Since we can't have a negative number of rows, we choose x = 5. Substituting this into the equation y = 2x + 4, we get y = 14. Therefore, there are 5 rows with 14 trees per row.

Question 20

The perimeter of a rectangle is 14 inches and the area is 12 square inches. Find the length and width of the rectangle.

Accepted Answer

Let the length of the rectangle be x and the width be y. We have two equations: 2x + 2y = 14 and xy = 12. Simplifying the first equation we get x + y = 7. Rearranging this equation we get y = 7 - x. Substituting in the second equation we get x(7 - x) = 12, which simplifies to x² - 7x + 12 = 0. Factoring this equation we get (x - 3)(x - 4) = 0, so x = 3 or x = 4. If x = 3, then y = 4 and if x = 4, then y = 3. The only option that matches these values is C. Therefore, the length is 4 inches and the width is 3 inches.

Quiz 8: A Transition From Elementary Algebra to Intermediate Algebra

Solve the equation using factoring techniques. 3 w 3 −-− 21 w 2 + 30 w = 0

The area of a triangular sheet of paper is 116 square inches. One side of the triangle is 5 inches more than three times the length of the altitude to that side. Find the length of that side and the altitude to the side.

Solve the equation using factoring techniques. w 2 −-− 3 w = 10 Please enter your answer as two numbers, separated by a comma.

Solve the equation using factoring techniques. x 2 + 8 x + 7 = 0 Please enter your answer as two numbers, separated by a comma.

Factor the polynomial completely. 9a2−24a+169 a ^ { 2 } - 24 a + 169a2−24a+16

Solve the equation using factoring techniques. 2 n 3 = 8 n Please enter your answer as three numbers, separated by commas.

Factor the polynomial completely. 9x2+169 x ^ { 2 } + 169x2+16

Solve the equation using factoring techniques. 3 t ( t −-− 7 ) = 0 Please enter your answer as two numbers, separated by a comma.

Factor the polynomial completely. 3x+3y+ax+ay3 x + 3 y + a x + a y3x+3y+ax+ay

Solve the equation using factoring techniques. ( x + 3 ) 2 −-− 25 = 0 Please enter your answer as two numbers, separated by a comma.

Factor the polynomial completely. x3−8x ^ { 3 } - 8x3−8

Solve the equation using factoring techniques. 5 w 3 −-− 60 w 2 + 175 w = 0 Please enter your answer as three numbers, separated by commas.

Factor the polynomial completely. 2n3+10n2−16n2 n ^ { 3 } + 10 n ^ { 2 } - 16 n2n3+10n2−16n