Question 1

Find the values of a, b, and c for which the quadratic equation ax2 + bx + c = 0 has the given numbers as solutions. Then
use those values to write a quadratic equation.
-$$- 2 , - 8$$

Accepted Answer

The sum of the solutions (-2 and -8) is equal to -b/a, and their product is c/a. Thus, -2 + (-8) = -10 and (-2)(-8) = 16. Therefore, a = 1, b = -10, and c = 16, leading to the equation $x^2 + 10x + 16 = 0$.

Question 2

Solve the problem.
-The length of a rectangle is 6 inches more than its width. If 3 inches are taken from the length and added to the width, the figure becomes a square with an area of 361 square inches. What are the
Dimensions of the original figure?

Accepted Answer

The answer of Solve the problem.
-The length of a rectangle...

Question 3

Answer the question.
-The mat around the picture shown measures x inches across. Which one of the following equations says that the area of the picture itself is 700 square inches?

Accepted Answer

The answer of Answer the question.
-The mat around the picture...

Question 4

Solve the problem.
-The area of a square is numerically 117 more than the perimeter. Find the length of the side.

Accepted Answer

The area of a square is given by $s^2$ and the perimeter by $4s$, where $s$ is the side length. The equation based on the given information is $s^2 = 4s + 117$. Rearranging, we get $s^2 - 4s - 117 = 0$. Factoring or using the quadratic formula, we find $s = 13$ as the positive solution that makes sense in this context.

Question 5

Solve the problem
-Find two consecutive odd integers whose product is 483.

Accepted Answer

The product of two consecutive odd integers can be expressed as $ n(n+2) = 483 $. Solving $ n^2 + 2n - 483 = 0 $ gives $ n = 21 $ or $ n = -23 $. Thus, the pairs are (21, 23) and (-21, -23).

Question 6

Solve the problem
-The sum of the squares of two consecutive even integers is 2452. Find the integers.

Accepted Answer

Let the two consecutive even integers be $x$ and $x+2$. Their squares are $x^2$ and $(x+2)^2$. The equation is $x^2 + (x+2)^2 = 2452$. Simplifying gives $2x^2 + 4x - 2448 = 0$, which simplifies to $x^2 + 2x - 1224 = 0$. Factoring or using the quadratic formula gives $x = 34$ or $x = -36$, so the integers are 34, 36 or -34, -36.

Question 7

Solve the problem
-The difference of the squares of two positive consecutive even integers is 68. Find the integers.

Accepted Answer

The answer of Solve the problem
-The difference of the squares...

Question 8

Find the values of a, b, and c for which the quadratic equation ax2 + bx + c = 0 has the given numbers as solutions. Then
use those values to write a quadratic equation.
-$$8 + 4 \sqrt { 2 } , 8 - 4 \sqrt { 2 }$$

Accepted Answer

Given the roots of the quadratic equation are $8 + 4\sqrt{2}$ and $8 - 4\sqrt{2}$, we can use the sum and product of roots formula for a quadratic equation $ax^2 + bx + c = 0$, where the sum of the roots $(-b/a)$ is $8 + 4\sqrt{2} + 8 - 4\sqrt{2} = 16$, and the product of the roots $(c/a)$ is $(8 + 4\sqrt{2})(8 - 4\sqrt{2}) = 64 - 32 = 32$. Therefore, the quadratic equation is $x^2 - 16x + 32 = 0$.

Question 9

Solve the problem.
-The outside of a picture frame has a length which is 3 cm more than width. The area enclosed by the outside of the picture frame is 130 square cm. Find the width of the outside of the picture frame.

Accepted Answer

The equation for the area of the rectangle is $ \text{length} \times \text{width} = \text{area} $. Given that the length is 3 cm more than the width, we can express the length as $ \text{width} + 3 $. Let $ w $ represent the width. The equation becomes $ (w + 3) \times w = 130 $. Simplifying, we get $ w^2 + 3w - 130 = 0 $. Factoring this quadratic equation, we find that $ w = 10 $ is the positive solution that makes sense in this context, as the width of a picture frame cannot be negative.

Question 10

Solve the problem
-Find two consecutive even integers whose product is 1520.

Accepted Answer

The equation for two consecutive even integers whose product is 1520 can be set up as $x(x + 2) = 1520$. Solving this quadratic equation, we find two sets of solutions: $x = 38$ and $x = -40$, which gives us the pairs (38, 40) and (-38, -40).

Question 11

Find the values of a, b, and c for which the quadratic equation ax2 + bx + c = 0 has the given numbers as solutions. Then
use those values to write a quadratic equation.
-$$- 4,8$$

Accepted Answer

The answer of Find the values of a, b, and...

Question 12

Solve the problem
-Find two consecutive integers whose product is 380. 252

Accepted Answer

The product of two consecutive integers that equals 380 can be both positive and negative pairs because the product of two negative numbers is positive. Thus, both 19 and 20, as well as -19 and -20, satisfy the condition.

Question 13

Find the values of a, b, and c for which the quadratic equation ax2 + bx + c = 0 has the given numbers as solutions. Then
use those values to write a quadratic equation.
-$$8 + \sqrt { 11 } , 8 - \sqrt { 11 }$$

Accepted Answer

The sum of the roots is $(8 + \sqrt{11}) + (8 - \sqrt{11}) = 16$, and the product of the roots is $(8 + \sqrt{11})(8 - \sqrt{11}) = 64 - 11 = 53$. For a quadratic equation $ax^2 + bx + c = 0$, the sum of the roots is $-b/a$ and the product is $c/a$. If $a = 1$, then $b = -16$ and $c = 53$, leading to the equation $x^2 - 16x + 53 = 0$.

Question 14

Find the values of a, b, and c for which the quadratic equation ax2 + bx + c = 0 has the given numbers as solutions. Then
use those values to write a quadratic equation.
-$4 i , - 4 i$

Accepted Answer

The solutions to a quadratic equation are given by the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. For complex solutions $4i$ and $-4i$, the sum of the solutions ($4i - 4i$) is 0, which means $b = 0$. The product of the solutions ($4i \cdot -4i$) is $-16i^2$, and since $i^2 = -1$, the product is 16. This corresponds to $c/a$, and since $a = 1$ for the quadratic equation in standard form $x^2 + bx + c = 0$, $c = 16$. Therefore, the correct equation is $x^2 + 16 = 0$.

Question 15

Find the values of a, b, and c for which the quadratic equation ax2 + bx + c = 0 has the given numbers as solutions. Then
use those values to write a quadratic equation.
-8,6

Accepted Answer

The sum of the roots (-b/a) is -8-6=-14, and the product of the roots (c/a) is 8*6=48. Thus, the correct equation is $x^2 - 14x + 48 = 0$.

Question 16

Solve the problem
-The sum of the squares of two consecutive odd integers is 1802. Find the integers.

Accepted Answer

The equation for the sum of the squares of two consecutive odd integers is $n^2 + (n+2)^2 = 1802$. Solving this equation, we find that $n = 29$ or $n = -31$, which means the consecutive odd integers can be 29 and 31 or -29 and -31.

Question 17

Solve the problem
-The sum of the squares of two consecutive integers is 221. Find the integers.

Accepted Answer

The equation for the sum of the squares of two consecutive integers $n$ and $n+1$ is $n^2 + (n+1)^2 = 221$. Simplifying, we get $2n^2 + 2n - 220 = 0$, which simplifies further to $n^2 + n - 110 = 0$. Factoring, we find $n = 10$ or $n = -11$. Therefore, the pairs are (10, 11) and (-11, -10), which corresponds to choice B.

Question 18

Find the values of a, b, and c for which the quadratic equation ax2 + bx + c = 0 has the given numbers as solutions. Then
use those values to write a quadratic equation.
-$16 i , - 16 i$

Accepted Answer

The sum of the roots is $16i - 16i = 0$, and the product of the roots is $(16i)(-16i) = -256i^2 = 256$. Thus, the quadratic equation is $x^2 - 0x + 256 = 0$ or $x^2 + 256 = 0$.

Question 19

Find the values of a, b, and c for which the quadratic equation ax2 + bx + c = 0 has the given numbers as solutions. Then
use those values to write a quadratic equation.
-$3 , - 9$

Accepted Answer

The solutions of the quadratic equation are 3 and -9, so the equation can be written as (x - 3)(x + 9) = 0, which expands to x^2 + 6x - 27 = 0.

Question 20

Solve the problem.
-The area of a square is numerically 4 less than the perimeter. Find the length of the side, if the side is greater than 1.

Accepted Answer

The answer of Solve the problem.
-The area of a square...

Quiz 2: Equations and Inequalities

Find the values of a, b, and c for which the quadratic equation ax2 + bx + c = 0 has the given numbers as solutions. Then use those values to write a quadratic equation. - $- 2 , - 8$

Solve the problem. -The length of a rectangle is 6 inches more than its width. If 3 inches are taken from the length and added to the width, the figure becomes a square with an area of 361 square inches. What are the Dimensions of the original figure?

Solve the problem. -The area of a square is numerically 117 more than the perimeter. Find the length of the side.

Solve the problem -Find two consecutive odd integers whose product is 483.

Solve the problem -The sum of the squares of two consecutive even integers is 2452. Find the integers.

Solve the problem -The difference of the squares of two positive consecutive even integers is 68. Find the integers.

Find the values of a, b, and c for which the quadratic equation ax2 + bx + c = 0 has the given numbers as solutions. Then use those values to write a quadratic equation. - $8 + 4 \sqrt { 2 } , 8 - 4 \sqrt { 2 }$

Solve the problem. -The outside of a picture frame has a length which is 3 cm more than width. The area enclosed by the outside of the picture frame is 130 square cm. Find the width of the outside of the picture frame.

Solve the problem -Find two consecutive even integers whose product is 1520.

Find the values of a, b, and c for which the quadratic equation ax2 + bx + c = 0 has the given numbers as solutions. Then use those values to write a quadratic equation. - $- 4,8$

Solve the problem -Find two consecutive integers whose product is 380. 252

Find the values of a, b, and c for which the quadratic equation ax2 + bx + c = 0 has the given numbers as solutions. Then use those values to write a quadratic equation. - $8 + \sqrt { 11 } , 8 - \sqrt { 11 }$

Find the values of a, b, and c for which the quadratic equation ax2 + bx + c = 0 has the given numbers as solutions. Then use those values to write a quadratic equation. - $4 i , - 4 i$

Find the values of a, b, and c for which the quadratic equation ax2 + bx + c = 0 has the given numbers as solutions. Then use those values to write a quadratic equation. -8,6

Solve the problem -The sum of the squares of two consecutive odd integers is 1802. Find the integers.

Solve the problem -The sum of the squares of two consecutive integers is 221. Find the integers.

Find the values of a, b, and c for which the quadratic equation ax2 + bx + c = 0 has the given numbers as solutions. Then use those values to write a quadratic equation. - $16 i , - 16 i$

Find the values of a, b, and c for which the quadratic equation ax2 + bx + c = 0 has the given numbers as solutions. Then use those values to write a quadratic equation. - $3 , - 9$

Solve the problem. -The area of a square is numerically 4 less than the perimeter. Find the length of the side, if the side is greater than 1.