Question
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Tutor-Verified Answer
Answer
\( 5x^2 + 28x + 15 = (x + 5)(5x + 3) \)
Solution
To factorize the quadratic expression \( 5x^2 + 28x + 15 \), follow these steps:
1. **Identify the coefficients:**
- \( a = 5 \) (coefficient of \( x^2 \))
- \( b = 28 \) (coefficient of \( x \))
- \( c = 15 \) (constant term)
2. **Find two numbers that multiply to \( a \times c = 5 \times 15 = 75 \) and add up to \( b = 28 \):**
- The numbers are **3** and **25** because:
- \( 3 \times 25 = 75 \)
- \( 3 + 25 = 28 \)
3. **Rewrite the middle term using these numbers:**
\[
5x^2 + 3x + 25x + 15
\]
4. **Factor by grouping:**
\[
(5x^2 + 3x) + (25x + 15)
\]
\[
x(5x + 3) + 5(5x + 3)
\]
5. **Factor out the common binomial factor \((5x + 3)\):**
\[
(x + 5)(5x + 3)
\]
**Final Factorized Form:**
\[
5x^2 + 28x + 15 = (x + 5)(5x + 3)
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
To factor the quadratic expression \( 5x^{2} + 28x + 15 \), we look for two numbers that multiply to \( 5 \times 15 = 75 \) and add up to \( 28 \). The numbers \( 25 \) and \( 3 \) fit the bill! Rewrite the expression as: \( 5x^{2} + 25x + 3x + 15 \). Now, group the terms: \( (5x^{2} + 25x) + (3x + 15) \). Factor out the common factors: \( 5x(x + 5) + 3(x + 5) \). Now, factor out \( (x + 5) \): \( (x + 5)(5x + 3) \). And voila! Your factored form is \( (x + 5)(5x + 3) \). Have fun checking by expanding to see that it retrieves the original expression!
