To factor the quadratic expression [tex]\(2x^2 - 14x + 24\)[/tex], we need to find its roots and then write the expression in its factored form.
1. Identify Coefficients:
The quadratic expression is in the form [tex]\(ax^2 + bx + c\)[/tex], where:
- [tex]\(a = 2\)[/tex]
- [tex]\(b = -14\)[/tex]
- [tex]\(c = 24\)[/tex]
2. Calculate the Discriminant:
The discriminant of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by [tex]\(\Delta = b^2 - 4ac\)[/tex].
[tex]\[
\Delta = (-14)^2 - 4(2)(24) = 196 - 192 = 4
\][/tex]
3. Find the Roots:
The roots of the quadratic equation can be found using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{\Delta}}{2a}\)[/tex].
[tex]\[
x = \frac{14 \pm \sqrt{4}}{4} = \frac{14 \pm 2}{4}
\][/tex]
This gives us two roots:
[tex]\[
x_1 = \frac{14 + 2}{4} = \frac{16}{4} = 4
\][/tex]
[tex]\[
x_2 = \frac{14 - 2}{4} = \frac{12}{4} = 3
\][/tex]
4. Write the Factored Form:
With the roots [tex]\(x_1 = 4\)[/tex] and [tex]\(x_2 = 3\)[/tex], the quadratic expression can be factored as:
[tex]\[
2(x - 4)(x - 3)
\][/tex]
So, the factored form of the expression [tex]\(2x^2 - 14x + 24\)[/tex] is [tex]\(2(x - 4)(x - 3)\)[/tex], which corresponds to option:
C. [tex]\(2(x - 3)(x - 4)\)[/tex]
(Note that since multiplication is commutative, both [tex]\(2(x-4)(x-3)\)[/tex] and [tex]\(2(x-3)(x-4)\)[/tex] are correct and equivalent.)
