Question
Question 5 Factor: \( 3 x^{2}+10 x+8 \) (A) \( (3 x+2)(x+4) \) (B) \( (3 x+4)(x+2) \) pry (C) \( (3 x+1)(x+8) \) (D) \( (3 x+2)(3 x+4) \)
Ask by Wagner Simmons. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The factored form of \(3x^2 + 10x + 8\) is \((3x + 4)(x + 2)\).
Solution
We are given the quadratic expression
\[
3x^2+10x+8.
\]
**Step 1.** Multiply the coefficient of \(x^2\) by the constant term:
\[
3 \times 8 = 24.
\]
**Step 2.** Find two numbers whose product is \(24\) and whose sum is \(10\). The numbers \(4\) and \(6\) satisfy:
\[
4 \times 6 = 24 \quad \text{and} \quad 4 + 6 = 10.
\]
**Step 3.** Rewrite the middle term \(10x\) using \(4x\) and \(6x\):
\[
3x^2 + 4x + 6x + 8.
\]
**Step 4.** Factor by grouping:
Group the first two terms and the last two terms:
\[
(3x^2 + 4x) + (6x + 8).
\]
Factor out the common factors from each group:
\[
x(3x + 4) + 2(3x + 4).
\]
**Step 5.** Notice that \((3x + 4)\) is common in both groups:
\[
(3x + 4)(x + 2).
\]
Thus, the factored form of the quadratic is:
\[
(3x + 4)(x + 2).
\]
The correct answer is option (B).
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Simplify this solution Bonus Knowledge
To factor the quadratic expression \( 3x^2 + 10x + 8 \), you can use the method of trial and error to determine which option correctly provides the product of the original polynomial. When you expand \( (3x + 4)(x + 2) \), you will find that it equals \( 3x^2 + 10x + 8 \). Thus, the correct answer is (B) \( (3x + 4)(x + 2) \). Looking at real-world applications, factoring quadratics isn't just for math class—it's also useful in various fields like physics and engineering! For example, when modeling projectile motion, the quadratic formula helps find the maximum height or time of flight, facilitating optimal design and real-life predictions.
