Question
17) \( x^{2}+x-30 \)
Ask by Garrett Schofield. in the United States
Mar 11,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( x^2 + x - 30 = (x + 6)(x - 5) \)
Solution
1. We start with the quadratic expression:
\[
x^2 + x - 30
\]
2. To factor the quadratic, we need to find two numbers that multiply to \(-30\) (the constant term) and add to \(1\) (the coefficient of \(x\)).
3. We examine the factor pairs of \(-30\):
- \(6\) and \(-5\):
\[
6 \cdot (-5) = -30 \quad \text{and} \quad 6 + (-5) = 1
\]
4. Since the numbers \(6\) and \(-5\) satisfy both conditions, we can write:
\[
x^2 + x - 30 = (x + 6)(x - 5)
\]
5. Thus, the factored form of the quadratic expression is:
\[
(x+6)(x-5)
\]
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Simplify this solution Extra Insights
Did you know that the quadratic equation \( x^{2}+x-30 \) can be factored to uncover its roots? By finding two numbers that multiply to \(-30\) (the constant term) and add to \(1\) (the coefficient of \(x\)), we discover that these numbers are \(6\) and \(-5\). Thus, the equation factors as \( (x + 6)(x - 5) = 0 \). You can solve for \(x\) by setting each factor to zero: \(x + 6 = 0\) gives you \(x = -6\), while \(x - 5 = 0\) gives you \(x = 5\). So, the solutions are \(x = -6\) and \(x = 5\). Now, you can visualize this as the points where a parabola crosses the x-axis—pretty neat, right?
