Question
solve for x factoring x^2 + 4x - 21 = 0
a. x = -3
b x = 3
c x= -7
d x=2
a. x = -3
b x = 3
c x= -7
d x=2
Answers
GPT-4o mini
To solve the quadratic equation \(x^2 + 4x - 21 = 0\) by factoring, we need to express it in the form \((x + m)(x + n) = 0\), where \(m\) and \(n\) are the roots.
1. First, we look for two numbers that multiply to \(-21\) (the constant term) and add to \(4\) (the coefficient of the \(x\) term).
2. The numbers that satisfy these conditions are \(7\) and \(-3\), because:
- \(7 \cdot (-3) = -21\)
- \(7 + (-3) = 4\)
Now, we can factor the quadratic:
\[
x^2 + 4x - 21 = (x + 7)(x - 3) = 0
\]
Next, we set each factor to zero:
1. \(x + 7 = 0 \Rightarrow x = -7\)
2. \(x - 3 = 0 \Rightarrow x = 3\)
So the solutions for \(x\) are \(x = -7\) and \(x = 3\).
From your options:
- a. \(x = -3\)
- b. \(x = 3\) ✔️
- c. \(x = -7\) ✔️
- d. \(x = 2\)
Both \(x = 3\) and \(x = -7\) are correct solutions.
1. First, we look for two numbers that multiply to \(-21\) (the constant term) and add to \(4\) (the coefficient of the \(x\) term).
2. The numbers that satisfy these conditions are \(7\) and \(-3\), because:
- \(7 \cdot (-3) = -21\)
- \(7 + (-3) = 4\)
Now, we can factor the quadratic:
\[
x^2 + 4x - 21 = (x + 7)(x - 3) = 0
\]
Next, we set each factor to zero:
1. \(x + 7 = 0 \Rightarrow x = -7\)
2. \(x - 3 = 0 \Rightarrow x = 3\)
So the solutions for \(x\) are \(x = -7\) and \(x = 3\).
From your options:
- a. \(x = -3\)
- b. \(x = 3\) ✔️
- c. \(x = -7\) ✔️
- d. \(x = 2\)
Both \(x = 3\) and \(x = -7\) are correct solutions.