The quadratic equation x2+4x−32=0

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

The numbers \(8\) and \(-4\) fit this requirement because:

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

Thus, the factored form of the given quadratic 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 find which of the given equations has exactly the same solutions as \((x-7)(x+9)=0\), we need to identify what solutions this equation has first.

Setting \((x-7)(x+9)=0\) gives us two solutions:

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

Now we will check each of the provided equations to see which has the same solutions:

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

   • Rearranging gives us \(x^2 + 2x - 63 = 0\), which can be factored as \((x - 7)(x + 9) = 0\). This has the same solutions: \(7\) and \(-9\).

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

   • This can be factored as \((x - 9)(x + 7) = 0\), which has solutions \(9\) and \(-7\). This does not have the same solutions.

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

   • Rearranging gives us \(x^2 - 2x + 63 = 0\), which does not have the same factorization as before.

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

   • Rearranging gives us \(x^2 + 2x + 63 = 0\), which also does not have the same factorization as before.

The only equation that simplifies to the same factors as the original equation is: \[ x^2 = -2x + 63 \]

Thus, the correct response is: \(x^2 = -2x + 63\).

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

−16
negative 16

16
16

−6
negative 6

6

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, which state that for a quadratic equation of the form \(x^2 + bx + c = 0\), the sum of the solutions \(-\frac{b}{a}\) and the product of the solutions \(\frac{c}{a}\).

For our equation:

1. The product of the solutions (\(-11\) and \(5\)) must equal \(c\) (in this case \(-55\)): \[ -11 \times 5 = -55 \] This is correct.

2. The sum of the solutions must equal \(-b\): \[ -11 + 5 = -6 \] Therefore: \[ -b = -6 \rightarrow b = 6 \]

Thus, the value of \(b\) should be \(6\).

The correct response is: 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=−64
x equals negative 64

x=64
x equals 64

x=−8

To solve the quadratic equation \(x^2 - 16x + 64 = 0\) using the perfect square trinomial pattern, we can rewrite the equation.

We notice that \(x^2 - 16x + 64\) can be expressed as: \[ (x - 8)^2 = 0 \]

This form follows the pattern \((a - b)^2 = a^2 - 2ab + b^2\). Here, \(a = x\) and \(b = 8\), since: \[ (x - 8)(x - 8) = x^2 - 8x - 8x + 64 = x^2 - 16x + 64 \]

Now we can set the equation to zero: \[ (x - 8)^2 = 0 \]

To solve for \(x\), take the square root of both sides: \[ x - 8 = 0 \]

Thus, solving for \(x\) gives us: \[ x = 8 \]

So the solution to the equation 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 \(x^2 - 36 = 0\) by factoring using the difference of squares pattern, we can recognize that the equation can be factored as follows:

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

Next, we can set each factor equal to zero to find the solutions:

1. \(x - 6 = 0\) gives: \[ x = 6 \]

2. \(x + 6 = 0\) gives: \[ x = -6 \]

Thus, the solutions to the equation \(x^2 - 36 = 0\) are: \[ x = 6 \quad \text{and} \quad x = -6 \]

Therefore, the correct response is: x = 6, x = -6.