In order to factor x ^ 2 – 2x – 24 we must break up the “bx” term into two terms whose coefficients…
add to-1 and multiply to -2.
add to 1 and multiply to -24.
add to-24 and multiply to -2.
add to -2 and multiply to -24.
The correct answer and explanation is:
The correct answer is “add to -2 and multiply to -24”.
To factor the quadratic expression x2−2x−24x^2 – 2x – 24, we start by looking at the form of a general quadratic equation, which is ax2+bx+cax^2 + bx + c. Here, a=1a = 1, b=−2b = -2, and c=−24c = -24.
Step 1: Identify the numbers
We need two numbers that:
- Add up to the coefficient of the middle term, which is b=−2b = -2.
- Multiply to the product of the coefficient of x2x^2 (which is 1) and the constant term c=−24c = -24. In this case, we need two numbers that multiply to 1×(−24)=−241 \times (-24) = -24.
Step 2: List possible pairs
Now, we look for pairs of numbers that multiply to -24 and check if their sum equals -2. The pairs of factors of -24 are:
- 1×(−24)=−241 \times (-24) = -24
- 2×(−12)=−242 \times (-12) = -24
- 3×(−8)=−243 \times (-8) = -24
- 4×(−6)=−244 \times (-6) = -24
- −1×24=−24-1 \times 24 = -24
- −2×12=−24-2 \times 12 = -24
- −3×8=−24-3 \times 8 = -24
- −4×6=−24-4 \times 6 = -24
The pair that adds up to -2 is −6-6 and 44, because: −6+4=−2-6 + 4 = -2
Step 3: Rewrite the middle term
Now, we can break up the middle term −2x-2x into two terms using −6x-6x and 4x4x: x2−6x+4x−24x^2 – 6x + 4x – 24
Step 4: Factor by grouping
Next, we factor by grouping. We group the first two terms and the last two terms: (x2−6x)+(4x−24)(x^2 – 6x) + (4x – 24)
Now, factor out the greatest common factor (GCF) from each group: x(x−6)+4(x−6)x(x – 6) + 4(x – 6)
Step 5: Final factorization
Finally, factor out the common binomial factor (x−6)(x – 6): (x+4)(x−6)(x + 4)(x – 6)
So, the factored form of x2−2x−24x^2 – 2x – 24 is (x+4)(x−6)(x + 4)(x – 6).
Thus, the correct pair of numbers is the one that adds to -2 and multiplies to -24.