The quadratic equation x^2+4x−32=0 can be solved by factoring. Which of the following is the factored form?(1 point) Responses (x+4)(x−8)=0 left parenthesis x plus 4 right parenthesis left parenthesis x minus 8 right parenthesis equals 0 (x−4)(x−8)=0 left parenthesis x minus 4 right parenthesis left parenthesis x minus 8 right parenthesis equals 0 (x+8)(x−4)=0 left parenthesis x plus 8 right parenthesis left parenthesis x minus 4 right parenthesis equals 0 (x+4)(x+8)=0

To factor the quadratic equation \( x^2 + 4x - 32 = 0 \), we need to find two numbers that multiply to \(-32\) (the constant term) and add up to \(4\) (the coefficient of \(x\)).

The two numbers that meet these criteria are \(8\) and \(-4\), because:

- \(8 \times -4 = -32\)
- \(8 + (-4) = 4\)

Thus, we can write the equation in its factored form:

\[
(x + 8)(x - 4) = 0
\]

So the correct answer is:

\[
(x + 8)(x - 4) = 0
\] (the third option).

Which of the following equations has exactly the same solutions as (x−7)(x+9)=0 ?(1 point)

x^2−2x−63=0
x^2=−2x+63
x^2=2x−63
x^2+2x=−63

To find which of the given equations has the same solutions as \( (x - 7)(x + 9) = 0 \), we first need to find the solutions to this equation.

Setting each factor equal to zero gives us:

1. \( x - 7 = 0 \) → \( x = 7 \)
2. \( x + 9 = 0 \) → \( x = -9 \)

Thus, the solutions to the equation \( (x - 7)(x + 9) = 0 \) are \( x = 7 \) and \( x = -9 \).

Now, let's check each equation to see if it has these same solutions:

1. **\( x^2 - 2x - 63 = 0 \)**

Factor this or use the quadratic formula:

\[
x^2 - 2x - 63 = 0 \quad \text{can be factored as} \quad (x - 9)(x + 7) = 0
\]

This gives the solutions \( x = 9 \) and \( x = -7 \), which do not match.

2. **\( x^2 = -2x + 63 \)**

Rearranging this gives:

\[
x^2 + 2x - 63 = 0
\]

Factor or use the quadratic formula:

\[
(x - 7)(x + 9) = 0
\]

This gives \( x = 7 \) and \( x = -9 \), which matches our initial equation.

3. **\( x^2 = 2x - 63 \)**

Rearranging gives:

\[
x^2 - 2x + 63 = 0
\]

The discriminant \( b^2 - 4ac = (-2)^2 - 4(1)(63) = 4 - 252 = -248 \) is negative, indicating no real solutions.

4. **\( x^2 + 2x = -63 \)**

Rearranging gives:

\[
x^2 + 2x + 63 = 0
\]

Again, the discriminant \( (2)^2 - 4(1)(63) = 4 - 252 = -248 \) is negative, indicating no real solutions.

Thus, the equation that has exactly the same solutions as \( (x−7)(x+9)=0 \) is:

\[
\boxed{x^2 = -2x + 63}
\]

What should be the value of b in x^2+bx−55=0 so that its solutions are −11 and 5? (1 point) Responses 6 6 −6 negative 6 16 16 −16

To find the value of \( b \) in the equation \( x^2 + bx - 55 = 0 \) such that its solutions are \( -11 \) and \( 5 \), we can use Vieta's formulas. According to Vieta's formulas, for a quadratic equation of the form \( x^2 + bx + c = 0 \):

1. The sum of the roots \( (-11 + 5) \) is equal to \( -b \).
2. The product of the roots \( (-11) \times (5) \) is equal to \( c \).

### Step 1: Calculate the sum of the roots
The sum of the roots is:
\[
-11 + 5 = -6
\]

According to Vieta's formulas, this is equal to \( -b \):
\[
-b = -6 \quad \Rightarrow \quad b = 6
\]

### Step 2: Verify the product of the roots
The product of the roots is:
\[
-11 \times 5 = -55
\]
This matches the constant term \( -55 \) in the equation.

Thus, the value of \( b \) that satisfies the given conditions is:

\[
\boxed{6}
\]

Solve the following quadratic equation using the perfect square trinomial pattern: x^2−16x+64=0 .(1 point) Responses x=8 x equals 8 x=−64 x equals negative 64 x=−8 x equals negative 8 x=64

To solve the quadratic equation \( x^2 - 16x + 64 = 0 \) using the perfect square trinomial pattern, we first recognize that this expression can be rewritten as:

\[
(x - 8)^2 = 0
\]

This is because:

\[
(x - 8)(x - 8) = x^2 - 16x + 64
\]

Now, we set the perfect square trinomial equal to zero:

\[
(x - 8)^2 = 0
\]

Next, we take the square root of both sides:

\[
x - 8 = 0
\]

Now, solving for \( x \):

\[
x = 8
\]

Thus, the solution to the equation \( x^2 - 16x + 64 = 0 \) is:

\[
\boxed{x = 8}
\]

Gabriel was asked to solve x^2−36=0 by factoring using the difference of squares pattern. What is/are the correct solution(s)?(1 point) Responses x=6 x equals 6 x=6 , x=−6 x equals 6 , x equals negative 6 x=18 , x=−18 x equals 18 , x equals negative 18 x=−6

The equation \( x^2 - 36 = 0 \) is a difference of squares, which can be factored as follows:

\[
x^2 - 36 = (x - 6)(x + 6) = 0
\]

Next, we set each factor equal to zero:

1. \( x - 6 = 0 \) → \( x = 6 \)
2. \( x + 6 = 0 \) → \( x = -6 \)

Therefore, the solutions to the equation \( x^2 - 36 = 0 \) are:

\[
x = 6 \quad \text{and} \quad x = -6
\]

So the correct answer is:

\[
\boxed{x = 6, x = -6}
\]