Question
solve for x by factoring x^2 + 4x - 21 = 0
Answers
GPT-4o mini
To solve the quadratic equation \( x^2 + 4x - 21 = 0 \) by factoring, we need to rewrite the equation in a product of two binomials.
We are looking for two numbers that multiply to \(-21\) (the constant term) and add to \(4\) (the coefficient of the \(x\) term).
The two numbers that satisfy these conditions are \(7\) and \(-3\) because:
- \(7 \times -3 = -21\)
- \(7 + (-3) = 4\)
Now, we can factor the quadratic:
\[
x^2 + 4x - 21 = (x + 7)(x - 3) = 0
\]
Next, we set each factor equal to zero:
1. \(x + 7 = 0\)
\[
x = -7
\]
2. \(x - 3 = 0\)
\[
x = 3
\]
Therefore, the solutions for the equation \( x^2 + 4x - 21 = 0 \) are:
\[
x = -7 \quad \text{and} \quad x = 3
\]
We are looking for two numbers that multiply to \(-21\) (the constant term) and add to \(4\) (the coefficient of the \(x\) term).
The two numbers that satisfy these conditions are \(7\) and \(-3\) because:
- \(7 \times -3 = -21\)
- \(7 + (-3) = 4\)
Now, we can factor the quadratic:
\[
x^2 + 4x - 21 = (x + 7)(x - 3) = 0
\]
Next, we set each factor equal to zero:
1. \(x + 7 = 0\)
\[
x = -7
\]
2. \(x - 3 = 0\)
\[
x = 3
\]
Therefore, the solutions for the equation \( x^2 + 4x - 21 = 0 \) are:
\[
x = -7 \quad \text{and} \quad x = 3
\]