Question

The quadratic equation x2+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 plus 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 left parenthesis x plus 4 right parenthesis left parenthesis x minus 8 right parenthesis equals 0 (x−4)(x−8)=0

Answer 1

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 to \(4\) (the coefficient of the linear term).

The numbers that satisfy this are \(8\) and \(-4\) because:

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

Thus, the factored form of the quadratic equation is:

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

So the correct response is:

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

Answer 2

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 x squared plus 2 x equals negative 63 x2−2x−63=0 x squared minus 2 x minus 63 equals 0 x2=2x−63 x squared equals 2 x minus 63

Answer 3

To find the equation that has exactly the same solutions as \((x - 7)(x + 9) = 0\), we first need to determine the solutions to this equation.

Setting each factor to zero gives us:

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

The solutions are \(x = 7\) and \(x = -9\).

Next, we can analyze the available options to rewrite them in standard form and check for the same roots.

Let's analyze the equations:

1. \(x^2 = -2x + 63\)
   Rearranging gives:
   \(x^2 + 2x - 63 = 0\)
   This could have the same solutions.

2. \(x^2 + 2x = -63\)
   Rearranging gives:
   \(x^2 + 2x + 63 = 0\)
   This does not have the same solutions (as the discriminant will be negative).

3. \(x^2 - 2x - 63 = 0\)
   This could have the same solutions.

4. \(x^2 = 2x - 63\)
   Rearranging gives:
   \(x^2 - 2x + 63 = 0\)
   This does not have the same solutions (as the discriminant will be negative).

5. \(x^2 = -2x + 63\)
   Rearranging gives:
   \(x^2 + 2x - 63 = 0\)
   This is the same as the first one.

6. \(x^2 = 2x - 63\)
   Rearranging gives:
   \(x^2 - 2x + 63 = 0\)
   This does not have the same solutions.

Checking the factors of \(x^2 + 2x - 63 = 0\) and \(x^2 - 2x - 63 = 0\):

• For \(x^2 + 2x - 63 = 0\):

The discriminant is \(2^2 - 4 \cdot 1 \cdot (-63) = 4 + 252 = 256\), and the roots are:
\[ x = \frac{-2 \pm 16}{2} \]
yielding \(x = 7\) and \(x = -9\).

• For \(x^2 - 2x - 63 = 0\):

The discriminant is \((-2)^2 - 4 \cdot 1 \cdot (-63) = 4 + 252 = 256\), and the roots are:
\[ x = \frac{2 \pm 16}{2} \]
yielding \(x = 9\) and \(x = -7\).

Conclusion:

The correct answer is:

1. \(x^2 = -2x + 63\) (which rearranges to \(x^2 + 2x - 63 = 0\)) or
2. \(x^2 - 2x - 63 = 0\) as they yield the same solutions of \(x = 7\) and \(x = -9\).

So, the equations \(x^2 = -2x + 63\) (first choice) and \(x^2 - 2x - 63 = 0\) (third choice) are both equivalent to \((x - 7)(x + 9) = 0\).

Answer 4

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

Answer 5

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 the relationship between the roots and the coefficients of a quadratic equation.

For a quadratic equation \( x^2 + bx + c = 0 \):

• The sum of the roots \( r_1 + r_2 = -b \)
• The product of the roots \( r_1 \cdot r_2 = c \)

Given the roots \( r_1 = -11 \) and \( r_2 = 5 \):

1. Calculate the sum of the roots: \[ r_1 + r_2 = -11 + 5 = -6 \] According to the relationship, this equals \(-b\): \[ -b = -6 \implies b = 6 \]

2. Check the product of the roots: \[ r_1 \cdot r_2 = -11 \cdot 5 = -55 \] This matches the constant term in the given equation.

Thus, the correct value of \( b \) is:

\[ \boxed{6} \]

Answer 6

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

Answer 7

To solve the quadratic equation \( x^2 - 16x + 64 = 0 \) using the perfect square trinomial pattern, we first observe that the left-hand side can be factored as follows:

The equation can be rewritten in the form of \( (x - a)^2 = 0 \).

In this case:

• The equation \( x^2 - 16x + 64 \) is a perfect square since: \[ x^2 - 16x + 64 = (x - 8)^2 \]

Thus, we can rewrite the equation as: \[ (x - 8)^2 = 0 \]

Now, we solve for \( x \) by taking the square root of both sides: \[ x - 8 = 0 \]

Solving for \( x \): \[ x = 8 \]

Therefore, the solution to the equation \( x^2 - 16x + 64 = 0 \) is: \[ \boxed{8} \]

Answer 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=18, x=−18 x equals 18 , x equals negative 18 x=6, x=−6 x equals 6 , x equals negative 6 x=−6 x equals negative 6 x=6

Answer 9

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

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

Now we can set each factor equal to zero:

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

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

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

So the correct responses are:

\[ \boxed{x = 6, , x = -6} \] or \[ \text{x equals 6, x equals negative 6} \]