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The quadratic equation x^2+4x−32=0 can be solved by factoring. Which of the following is the factored form?

Responses:

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

(x−4)(x−8)=0

(x+4)(x−8)=0

(x+8)(x−4)=0

9 answers

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 these conditions are \(8\) and \(-4\), since:

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

So, we can factor the quadratic as:

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

Thus, the factored form of the equation is:

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

Therefore, 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

Responses

x^2=−2x+63

x^2=2x−63

x^2+2x=−63

x^2−2x−63=0

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

Setting each factor to zero gives us the solutions:

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

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

Now, we will check each response to see which one is equivalent to the factors we have.

  1. For \(x^2 = -2x + 63\): \[ x^2 + 2x - 63 = 0 \quad \text{(add \(2x\) and subtract \(63\))} \]

    We can factor this: \[ (x - 7)(x + 9) = 0 \] This has the same solutions.

  2. For \(x^2 = 2x - 63\): \[ x^2 - 2x + 63 = 0 \] This does not factor to yield our original solutions.

  3. For \(x^2 + 2x = -63\): \[ x^2 + 2x + 63 = 0 \] This does not factor to yield our original solutions.

  4. For \(x^2 - 2x - 63 = 0\): This factors as: \[ (x - 9)(x + 7) = 0 \] This also gives different solutions.

Based on this analysis, the only equation that has the same solutions \(x = 7\) and \(x = -9\) is:

x^2 = -2x + 63

What should be the value of b in x^2+bx−55=0 so that its solutions are −11and 5?
Responses:

−16

6

−6

16

To determine the value of \(b\) in the quadratic equation \(x^2 + bx - 55 = 0\) given that the solutions are \(-11\) and \(5\), we can use Vieta's formulas, which relate the coefficients of the polynomial to sums and products of its roots.

For a quadratic equation of the form \(x^2 + bx + c = 0\), if the roots are \(r_1\) and \(r_2\):

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

In this case:

  • Roots: \(r_1 = -11\) and \(r_2 = 5\)
  • The product of the roots: \[ r_1 \cdot r_2 = -11 \cdot 5 = -55 \]
  • The sum of the roots: \[ r_1 + r_2 = -11 + 5 = -6 \]

By Vieta's, this means: \[-b = -6\]

Thus, we can find \(b\): \[ b = 6 \]

So, the value of \(b\) should be:

6

Solve the following quadratic equation using the perfect square trinomial pattern: x^2−16x+64=0

Responses

x=−8

x=−64

x=8

x=64

To solve the quadratic equation \(x^2 - 16x + 64 = 0\) using the perfect square trinomial pattern, we first need to recognize that the left side of the equation can be factored as a perfect square.

The expression \(x^2 - 16x + 64\) can be written as:

\[ (x - 8)^2 \]

This is because: \[ (x - 8)(x - 8) = x^2 - 8x - 8x + 64 = x^2 - 16x + 64 \]

So, the equation can be rewritten as:

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

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

\[ x - 8 = 0 \]

Now, we solve for \(x\):

\[ x = 8 \]

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

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)?
Responses

x=6, x=−6

x=18, x=−18

x=−6

x=6

The equation \(x^2 - 36 = 0\) can be factored using the difference of squares pattern. The difference of squares states that:

\[ a^2 - b^2 = (a - b)(a + b) \]

In this case, \(x^2 - 36\) can be expressed as:

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

Setting that equal to zero gives:

\[ (x - 6)(x + 6) = 0 \]

Now we set each factor equal to zero:

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

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