The quadratic equation x2+4x−32=0 can be solved by factoring. Which of the following is the factored form?(1 point) Responses (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 left parenthesis x plus 4 right parenthesis left parenthesis x plus 8 right parenthesis equals 0 (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

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 the middle term).

The two numbers that satisfy these conditions are \(8\) and \(-4\).

Therefore, the factored form of the equation is:

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

So the correct response is:

(x + 8)(x - 4) = 0.

Which of the following equations has exactly the same solutions as (x−7)(x+9)=0 ?(1 point) Responses x2=−2x+63 x squared equals negative 2 x plus 63 x2−2x−63=0 x squared minus 2 x minus 63 equals 0 x2=2x−63 x squared equals 2 x minus 63 x2+2x=−63

To solve the equation \((x - 7)(x + 9) = 0\), we can find the solutions by setting each factor equal to zero:

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

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

Now let's analyze the options given:

1. \(x^2 = -2x + 63\)
   Rearranging gives \(x^2 + 2x - 63 = 0\).

2. \(x^2 - 2x - 63 = 0\)
   This can be factored or solved separately.

3. \(x^2 = 2x - 63\)
   Rearranging gives \(x^2 - 2x + 63 = 0\) which does not match.

4. \(x^2 + 2x = -63\)
   Rearranging gives \(x^2 + 2x + 63 = 0\) which does not match.

Next, we can check option 1 and 2:

1. The quadratic \(x^2 + 2x - 63 = 0\) can be factored into \((x - 7)(x + 9) = 0\), which has the same solutions \(x = 7\) and \(x = -9\).

2. The quadratic \(x^2 - 2x - 63 = 0\) factors into \((x - 9)(x + 7) = 0\), which has the solutions \(x = 9\) and \(x = -7\), but these are not the same.

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

\(x^2 = -2x + 63\), which rearranges to \(x^2 + 2x - 63 = 0\).

What should be the value of b in x2+bx−55=0 so that its solutions are −11 and 5? (1 point) Responses −6 negative 6 −16 negative 16 6 6 16

To find the value of \( b \) in the quadratic 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 \( x^2 + bx + c = 0 \):

• The sum of the solutions (roots) \( r_1 + r_2 \) is equal to \( -b \).
• The product of the solutions \( r_1 \times r_2 \) is equal to \( c \).

Given the solutions are \( -11 \) and \( 5 \):

1. Sum of the solutions: \[ -11 + 5 = -6 \] Thus, we have: \[ -b = -6 \implies b = 6 \]

2. Product of the solutions (just to verify): \[ (-11) \times 5 = -55 \] This matches the constant term of the equation, confirming that the product is correct.

Therefore, the value of \( b \) is:

6.

u sure?

Let’s double-check the calculations to ensure we have the correct value of \( b \) in the equation \( x^2 + bx - 55 = 0 \) given the roots \( -11 \) and \( 5 \).

1. Sum of the roots: According to Vieta's formulas, the sum of the roots \( r_1 \) and \( r_2 \) is related to the coefficient \( b \) as follows: \[ r_1 + r_2 = -b \] So: \[ -11 + 5 = -6 \] This means: \[ -b = -6 \implies b = 6 \]

2. Product of the roots: The product of the roots should equal the constant term: \[ r_1 \times r_2 = -55 \] Calculate: \[ (-11) \times 5 = -55 \] This is correct.

From these calculations, we confirm:

• The sum of the roots gives \( b = 6\).
• The product of the roots verifies the constant term.

So yes, the value of \( b \) is indeed 6.

Solve the following quadratic equation using the perfect square trinomial pattern: x2−16x+64=0 .(1 point) Responses x=64 x equals 64 x=−8 x equals negative 8 x=−64 x equals negative 64 x=8

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

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

This is because:

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

Now, we can set the equation equal to zero:

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

To solve for \( x \), we take the square root of both sides:

\[ x - 8 = 0 \]

Solving for \( x \):

\[ x = 8 \]

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

x = 8.

Gabriel was asked to solve x2−36=0 by factoring using the difference of squares pattern. What is/are the correct solution(s)?(1 point) Responses x=−6 x equals negative 6 x=18 , x=−18 x equals 18 , x equals negative 18 x=6 , x=−6 x equals 6 , x equals negative 6 x=6

To solve the equation \( x^2 - 36 = 0 \) using the difference of squares pattern, we can recognize that it can be factored as follows:

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

Setting each factor equal to zero gives us the solutions:

1. \( x - 6 = 0 \) which leads to \( x = 6 \)
2. \( x + 6 = 0 \) which leads to \( x = -6 \)

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

x = 6 and x = -6.

So the correct response is:

x = 6, x = -6.