Which equation is set up for direct use of the zero-factor property? Solve it.
A. 3 x squared minus 17 x minus 6 equals 0
B. left parenthesis 5 x plus 6 right parenthesis squared equals 7
C. x squared plus x equals 6
D. left parenthesis 3 x minus 1 right parenthesis left parenthesis x minus 4 right parenthesis equals 0
The correct answer and explanation is:
The equation that is set up for direct use of the zero-factor property is D: (3x−1)(x−4)=0(3x – 1)(x – 4) = 0
Explanation:
The Zero-Product Property states that if the product of two expressions equals zero, then at least one of the expressions must be zero. Mathematically, if: (a)(b)=0(a)(b) = 0
then either a=0a = 0 or b=0b = 0.
In equation D, the product of (3x−1)(3x – 1) and (x−4)(x – 4) is set equal to zero. To use the zero-product property, we can set each factor equal to zero individually: 3x−1=0orx−4=03x – 1 = 0 \quad \text{or} \quad x – 4 = 0
Solving each equation:
- For 3x−1=03x – 1 = 0: 3x=13x = 1 x=13x = \frac{1}{3}
- For x−4=0x – 4 = 0: x=4x = 4
So the solutions to the equation (3x−1)(x−4)=0(3x – 1)(x – 4) = 0 are x=13x = \frac{1}{3} and x=4x = 4.
Why the other options don’t fit the zero-factor property:
- Option A (3×2−17x−6=0)(3x^2 – 17x – 6 = 0): This is a quadratic equation. Although it could potentially be factored, it is not set up as the product of two factors equaling zero, which is necessary for the zero-product property.
- Option B (5x+6)2=7(5x + 6)^2 = 7: This equation involves a square, not a product of two terms set equal to zero. It would require other methods, such as isolating xx, to solve.
- Option C (x2+x=6)(x^2 + x = 6): This equation would need to be rearranged into a quadratic form (x2+x−6=0x^2 + x – 6 = 0) before it could be factored and solved, but it’s not directly in the form for zero-factor property use.
Hence, the correct choice is D: (3x−1)(x−4)=0(3x – 1)(x – 4) = 0.