Question
Use factoring to solve the quadratic equation. Check by \( x^{2}-5 x-14=0 \) Rewrite the equation in factored form. \( (x-7)(x+2)=0 \) (Factor completely.) The solution set is \( \{\square\} \) (Use a comma to separate answers as needed. Type
Ask by Schwartz Coleman. in the United States
Jan 27,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The solutions are \( x = 7 \) and \( x = -2 \).
Solution
To solve the quadratic equation \( x^{2} - 5x - 14 = 0 \) by factoring, follow these steps:
1. **Factor the quadratic:**
The equation factors to:
\[
(x - 7)(x + 2) = 0
\]
2. **Set each factor equal to zero and solve for \( x \):**
\[
x - 7 = 0 \quad \Rightarrow \quad x = 7
\]
\[
x + 2 = 0 \quad \Rightarrow \quad x = -2
\]
3. **Write the solution set:**
\[
\{7, -2\}
\]
**Verification:**
Plugging the solutions back into the original equation to verify:
- For \( x = 7 \):
\[
(7)^2 - 5(7) - 14 = 49 - 35 - 14 = 0
\]
- For \( x = -2 \):
\[
(-2)^2 - 5(-2) - 14 = 4 + 10 - 14 = 0
\]
Both solutions satisfy the original equation.
**Final Answer:**
\(\{7, -2\}\)
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
To factor the quadratic equation \( x^{2} - 5x - 14 = 0 \), you can look for two numbers that multiply to \(-14\) (the constant term) and add up to \(-5\) (the coefficient of \(x\)). Those numbers are \(-7\) and \(2\), allowing you to rewrite the equation as \((x - 7)(x + 2) = 0\). To find the solution set, set each factor equal to zero. Solving \(x - 7 = 0\) gives \(x = 7\), and solving \(x + 2 = 0\) provides \(x = -2\). Thus, the solution set is \(\{7, -2\}\).
