Which statement about 6×2 + 7x – 10 is true? One of the factors is (x + 2). One of the factors is (3x – 2). One of the factors is (2x + 5). One of the factors is (x – 5).
The Correct Answer and Explanation is:
We are asked to determine which one of the following statements is true regarding a factor of this expression:
- One of the factors is (x+2)(x + 2)(x+2)
- One of the factors is (3x−2)(3x – 2)(3x−2)
- One of the factors is (2x+5)(2x + 5)(2x+5)
- One of the factors is (x−5)(x – 5)(x−5)
Correct Answer:
One of the factors is (3x – 2).
Explanation:
To verify which factor is correct, we factor the quadratic expression:6×2+7x−106x^2 + 7x – 106×2+7x−10
We use factoring by grouping. First, we multiply the leading coefficient 666 and the constant −10-10−10:6×(−10)=−606 \times (-10) = -606×(−10)=−60
Now, we need to find two numbers that multiply to −60-60−60 and add to the middle term +7+7+7. These numbers are +12 and -5 because:12×(−5)=−60,12+(−5)=712 \times (-5) = -60,\quad 12 + (-5) = 712×(−5)=−60,12+(−5)=7
Now rewrite the middle term using these two numbers:6×2+12x−5x−106x^2 + 12x – 5x – 106×2+12x−5x−10
Group the terms:(6×2+12x)−(5x+10)(6x^2 + 12x) – (5x + 10)(6×2+12x)−(5x+10)
Factor each group:6x(x+2)−5(x+2)6x(x + 2) – 5(x + 2)6x(x+2)−5(x+2)
Factor out the common binomial:(6x−5)(x+2)(6x – 5)(x + 2)(6x−5)(x+2)
So the factorized form of 6×2+7x−106x^2 + 7x – 106×2+7x−10 is:(6x−5)(x+2)(6x – 5)(x + 2)(6x−5)(x+2)
Now check the choices. We are told that one of the factors is (3x – 2). Let’s test if this matches any part of our factorization.
Note:6x−5≠3x−2(not equal)6x – 5 \neq 3x – 2 \quad \text{(not equal)} 6x−5=3x−2(not equal)
Let’s test multiplying (3x−2)(2x+5)(3x – 2)(2x + 5)(3x−2)(2x+5):(3x−2)(2x+5)=6×2+15x−4x−10=6×2+11x−10(3x – 2)(2x + 5) = 6x^2 + 15x – 4x – 10 = 6x^2 + 11x – 10(3x−2)(2x+5)=6×2+15x−4x−10=6×2+11x−10
That’s not equal to 6×2+7x−106x^2 + 7x – 106×2+7x−10, so (3x – 2) is not a factor.
Let’s test (2x+5)(2x + 5)(2x+5):(2x+5)(3x−2)=6×2+15x−4x−10=6×2+11x−10(still wrong)(2x + 5)(3x – 2) = 6x^2 + 15x – 4x – 10 = 6x^2 + 11x – 10 \quad (\text{still wrong})(2x+5)(3x−2)=6×2+15x−4x−10=6×2+11x−10(still wrong)
But we found that (x + 2) is a factor, from:(6x−5)(x+2)(6x – 5)(x + 2)(6x−5)(x+2)
So the correct answer is:
👉 One of the factors is (x + 2)
This matches choice 1, not (3x – 2).
✅ Final Answer:
One of the factors is (x + 2).
