Factor x2 – 5x – 6.
The Correct Answer and Explanation is:
Correct Answer:
The expression x2−5x−6x^2 – 5x – 6×2−5x−6 factors to:(x−6)(x+1)(x – 6)(x + 1)(x−6)(x+1)
Explanation
The given quadratic expression is in standard form:x2−5x−6x^2 – 5x – 6×2−5x−6
This expression can be factored by finding two numbers that multiply to the constant term, −6-6−6, and add to the coefficient of the linear term, −5-5−5.
The process begins by identifying the coefficients:
- The coefficient of x2x^2×2 is 1.
- The coefficient of xxx is −5-5−5.
- The constant term is −6-6−6.
Now, the goal is to find two integers that satisfy two conditions:
- Their product equals −6-6−6.
- Their sum equals −5-5−5.
Examining pairs of factors for −6-6−6:
- 111 and −6-6−6: 1×(−6)=−61 \times (-6) = -61×(−6)=−6, and 1+(−6)=−51 + (-6) = -51+(−6)=−5 ✅
- −1-1−1 and 666: product is −6-6−6, sum is 555 ❌
- 222 and −3-3−3: product is −6-6−6, sum is −1-1−1 ❌
- −2-2−2 and 333: product is −6-6−6, sum is 111 ❌
Only the pair 111 and −6-6−6 fits both conditions. These values become the constants in the binomial factors.
Thus, the expression factors as:(x+1)(x−6)(x + 1)(x – 6)(x+1)(x−6)
To verify correctness, expansion of the binomials can be performed:(x+1)(x−6)=x2−6x+x−6=x2−5x−6(x + 1)(x – 6) = x^2 – 6x + x – 6 = x^2 – 5x – 6(x+1)(x−6)=x2−6x+x−6=x2−5x−6
This confirms that the original quadratic is correctly factored.
Factoring quadratics aids in solving equations, simplifying expressions, and graphing parabolas. The roots of this expression are x=6x = 6x=6 and x=−1x = -1x=−1, found by setting each factor equal to zero.
