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.
Here’s the explanation:
When factoring a quadratic expression in the form of x2+bx+cx^2 + bx + c, the goal is to find two numbers that add to the coefficient of the xx-term (which is bb) and multiply to the constant term (which is cc).
For the given quadratic expression x2−2x−24x^2 – 2x – 24, we need to break the middle term −2x-2x into two terms whose coefficients:
- Add to −2-2 (the coefficient of xx),
- Multiply to −24-24 (the constant term).
We are looking for two numbers, say mm and nn, that satisfy these two conditions:
- m+n=−2m + n = -2 (the sum of the numbers must be equal to −2-2),
- m×n=−24m \times n = -24 (the product of the numbers must be equal to −24-24).
By examining possible factor pairs of −24-24, we get:
- 1×−241 \times -24,
- −1×24-1 \times 24,
- 2×−122 \times -12,
- −2×12-2 \times 12,
- 3×−83 \times -8,
- −3×8-3 \times 8,
- 4×−64 \times -6,
- −4×6-4 \times 6.
Out of these, the pair that adds up to −2-2 is 44 and −6-6. That is:
- 4+(−6)=−24 + (-6) = -2,
- 4×(−6)=−244 \times (-6) = -24.
So, we can rewrite −2x-2x as 4x−6x4x – 6x, which gives the expression: x2+4x−6x−24.x^2 + 4x – 6x – 24.
Next, we factor by grouping: x(x+4)−6(x+4).x(x + 4) – 6(x + 4).
Now, we factor out the common binomial (x+4)(x + 4): (x−6)(x+4).(x – 6)(x + 4).
Thus, the factored form of x2−2x−24x^2 – 2x – 24 is (x−6)(x+4)(x – 6)(x + 4).