If 4x – 3 is a factor of 8x^2 + 14x – 15, find the other factor?
The Correct Answer and Explanation is:
Correct Answer:
The other factor is 2x + 5
Step-by-step Explanation (Without Using the Word “We”):
Given that 4x – 3 is a factor of the quadratic expression:8×2+14x−158x^2 + 14x – 158×2+14x−15
The goal is to find the other factor.
Start by assuming the product of two binomials equals the expression. Since one factor is already known:8×2+14x−15=(4x−3)(ax+b)8x^2 + 14x – 15 = (4x – 3)(ax + b)8×2+14x−15=(4x−3)(ax+b)
To find the values of a and b, expand the right-hand side:(4x−3)(ax+b)=4ax2+(4b−3a)x−3b(4x – 3)(ax + b) = 4a x^2 + (4b – 3a)x – 3b(4x−3)(ax+b)=4ax2+(4b−3a)x−3b
Now match this with the original quadratic:8×2+14x−15=4ax2+(4b−3a)x−3b8x^2 + 14x – 15 = 4a x^2 + (4b – 3a)x – 3b8x2+14x−15=4ax2+(4b−3a)x−3b
Match the coefficients on both sides:
- 4a=8⇒a=24a = 8 \Rightarrow a = 24a=8⇒a=2
- −3b=−15⇒b=5-3b = -15 \Rightarrow b = 5−3b=−15⇒b=5
- Verify middle term: 4b−3a=4(5)−3(2)=20−6=144b – 3a = 4(5) – 3(2) = 20 – 6 = 144b−3a=4(5)−3(2)=20−6=14
All terms match the original expression, confirming correctness.
Therefore, the full factorization of the quadratic is:8×2+14x−15=(4x−3)(2x+5)8x^2 + 14x – 15 = (4x – 3)(2x + 5)8×2+14x−15=(4x−3)(2x+5)
So, the other factor is:2x+5\boxed{2x + 5}2x+5
Summary:
The quadratic expression was factored using expansion and coefficient comparison. Since the known factor was 4x−34x – 34x−3, matching the resulting expression with the original quadratic led to determining the unknown binomial. All coefficients aligned perfectly, verifying the solution. Hence, 2x+52x + 52x+5 is the other factor.
