Solving Quadratic Equations Using Patterns Quick Check

Question

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

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
x squared equals negative 2 x plus 63

x2+2x=−63
x squared plus 2 x equals negative 63
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Answer 1

To determine which equation has exactly the same solutions as \((x−7)(x+9)=0\), we first find the solutions of the original equation.

The equation \((x−7)(x+9)=0\) implies:

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

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

Next, we will analyze each of the given equations to see if they produce the same solutions:

1. \(x^2 - 2x - 63 = 0\)
   We can factor this equation or use the quadratic formula: \[ x^2 - 2x - 63 = (x-9)(x+7) = 0 \] The solutions are \(x = 9\) and \(x = -7\) (not the same as \(x = 7\) and \(x = -9\)).

2. \(x^2 = 2x - 63\)
   Rearranging gives: \[ x^2 - 2x + 63 = 0 \] Using the quadratic formula: \[ x = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot 63}}{2 \cdot 1} = \frac{2 \pm \sqrt{4 - 252}}{2} = \frac{2 \pm \sqrt{-248}}{2} \] This will yield complex solutions (not the same).

3. \(x^2 = -2x + 63\)
   Rearranging gives: \[ x^2 + 2x - 63 = 0 \] Factoring or using the quadratic formula: \[ x^2 + 2x - 63 = (x - 7)(x + 9) = 0 \] The solutions are \(x = 7\) and \(x = -9\) (which are exactly the same as the original equation).

4. \(x^2 + 2x = -63\)
   Rearranging gives: \[ x^2 + 2x + 63 = 0 \] This will also yield complex solutions (not the same).

The correct option that has the same solutions as \((x−7)(x+9)=0\) is:

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