To solve the equation [tex]\(4x^2 - 7x = 3x + 24\)[/tex], we will follow these steps:
1. Move all terms to one side of the equation to set it to zero:
[tex]\[
4x^2 - 7x - 3x - 24 = 0
\][/tex]
2. Combine like terms:
[tex]\[
4x^2 - 10x - 24 = 0
\][/tex]
3. Find the roots of the quadratic equation [tex]\(4x^2 - 10x - 24 = 0\)[/tex]. To do this, we use the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\(a = 4\)[/tex], [tex]\(b = -10\)[/tex], and [tex]\(c = -24\)[/tex].
- Calculate the discriminant:
[tex]\[
b^2 - 4ac = (-10)^2 - 4(4)(-24) = 100 + 384 = 484
\][/tex]
- Compute the roots using the quadratic formula:
[tex]\[
x = \frac{-(-10) \pm \sqrt{484}}{2 \cdot 4} = \frac{10 \pm 22}{8}
\][/tex]
4. Simplify the solutions:
[tex]\[
x = \frac{10 + 22}{8} = \frac{32}{8} = 4
\][/tex]
[tex]\[
x = \frac{10 - 22}{8} = \frac{-12}{8} = -\frac{3}{2}
\][/tex]
The solutions to the quadratic equation [tex]\(4x^2 - 10x - 24 = 0\)[/tex] are [tex]\(x = 4\)[/tex] and [tex]\(x = -\frac{3}{2}\)[/tex].
5. Checking which solutions apply from the given options:
- [tex]\(x = -4\)[/tex] does not apply.
- [tex]\(x = -3\)[/tex] does not apply.
- [tex]\(x = -\frac{3}{2}\)[/tex] applies.
- [tex]\(x = \frac{2}{3}\)[/tex] does not apply.
- [tex]\(x = 2\)[/tex] does not apply.
- [tex]\(x = 4\)[/tex] applies.
Therefore, the solutions to the equation [tex]\(4x^2 - 7x = 3x + 24\)[/tex] are:
- [tex]\(x = -\frac{3}{2}\)[/tex]
- [tex]\(x = 4\)[/tex]
