Let's solve the equation [tex]\(4x^2 - 7x = 3x + 24\)[/tex] step-by-step to find its solutions.
1. Start with the given equation:
[tex]\[
4x^2 - 7x = 3x + 24
\][/tex]
2. Move all terms to one side to set the equation to 0:
[tex]\[
4x^2 - 7x - 3x - 24 = 0
\][/tex]
Simplify the equation:
[tex]\[
4x^2 - 10x - 24 = 0
\][/tex]
3. Use the quadratic formula to solve the simplified equation [tex]\(4x^2 - 10x - 24 = 0\)[/tex]:
The quadratic formula is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, the coefficients are [tex]\(a = 4\)[/tex], [tex]\(b = -10\)[/tex], and [tex]\(c = -24\)[/tex].
4. Calculate the discriminant:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
So,
[tex]\[
\Delta = (-10)^2 - 4 \cdot 4 \cdot (-24)
\][/tex]
[tex]\[
\Delta = 100 + 384
\][/tex]
[tex]\[
\Delta = 484
\][/tex]
5. Take the square root of the discriminant:
[tex]\[
\sqrt{484} = 22
\][/tex]
6. Plug the values into the quadratic formula:
[tex]\[
x = \frac{-(-10) \pm 22}{2 \cdot 4}
\][/tex]
Simplify:
[tex]\[
x = \frac{10 \pm 22}{8}
\][/tex]
7. Calculate the two possible solutions:
- For the positive root:
[tex]\[
x = \frac{10 + 22}{8} = \frac{32}{8} = 4
\][/tex]
- For the negative root:
[tex]\[
x = \frac{10 - 22}{8} = \frac{-12}{8} = -\frac{3}{2}
\][/tex]
So, the solutions to the equation [tex]\(4x^2 - 10x - 24 = 0\)[/tex] are:
[tex]\[
x = -\frac{3}{2} \quad \text{and} \quad x = 4
\][/tex]
8. Check which of the given options match these solutions:
- [tex]\( x = -4 \)[/tex] is not a solution.
- [tex]\( x = -3 \)[/tex] is not a solution.
- [tex]\( x = -\frac{3}{2} \)[/tex] is a solution.
- [tex]\( x = \frac{2}{3} \)[/tex] is not a solution.
- [tex]\( x = 2 \)[/tex] is not a solution.
- [tex]\( x = 4 \)[/tex] is a solution.
Therefore, the correct solutions are:
[tex]\[
x = -\frac{3}{2} \quad \text{and} \quad x = 4
\][/tex]
