Question
The Solving Quadratic Equations Using Patterns Quick Check
which of the following equations has exactly the same solutions as (x-7)(x+9)=0
which of the following equations has exactly the same solutions as (x-7)(x+9)=0
Answers
GPT-4o mini
To find which equation has exactly the same solutions as \((x - 7)(x + 9) = 0\), we first need to identify the solutions of the given equation.
1. Set each factor to zero:
- \(x - 7 = 0\) gives \(x = 7\).
- \(x + 9 = 0\) gives \(x = -9\).
Thus, the solutions of the equation \((x - 7)(x + 9) = 0\) are \(x = 7\) and \(x = -9\).
Now, we can write a new equation using these solutions. The factors corresponding to the solutions can be expressed as:
\[
(x - 7)(x + 9) = 0.
\]
The original equation is already in factored form. However, we can also express it in standard quadratic form by expanding it:
\[
x^2 + 9x - 7x - 63 = 0,
\]
which simplifies to:
\[
x^2 + 2x - 63 = 0.
\]
Any equation that can be expressed in the form of \((x - 7)(x + 9) = 0\) or that is equivalent to \(x^2 + 2x - 63 = 0\) will have the same solutions \(x = 7\) and \(x = -9\).
To summarize, any equation that is equivalent to \((x - 7)(x + 9) = 0\) or has the same roots will also have exactly these same solutions. Possible equivalent forms include:
- \((x - 7)(x + 9) = 0\)
- \(x^2 + 2x - 63 = 0\)
To check other equations, you would factor them or compute their roots to see if they yield \(x = 7\) and \(x = -9\) as solutions.
1. Set each factor to zero:
- \(x - 7 = 0\) gives \(x = 7\).
- \(x + 9 = 0\) gives \(x = -9\).
Thus, the solutions of the equation \((x - 7)(x + 9) = 0\) are \(x = 7\) and \(x = -9\).
Now, we can write a new equation using these solutions. The factors corresponding to the solutions can be expressed as:
\[
(x - 7)(x + 9) = 0.
\]
The original equation is already in factored form. However, we can also express it in standard quadratic form by expanding it:
\[
x^2 + 9x - 7x - 63 = 0,
\]
which simplifies to:
\[
x^2 + 2x - 63 = 0.
\]
Any equation that can be expressed in the form of \((x - 7)(x + 9) = 0\) or that is equivalent to \(x^2 + 2x - 63 = 0\) will have the same solutions \(x = 7\) and \(x = -9\).
To summarize, any equation that is equivalent to \((x - 7)(x + 9) = 0\) or has the same roots will also have exactly these same solutions. Possible equivalent forms include:
- \((x - 7)(x + 9) = 0\)
- \(x^2 + 2x - 63 = 0\)
To check other equations, you would factor them or compute their roots to see if they yield \(x = 7\) and \(x = -9\) as solutions.