To solve the quadratic equation [tex]\(4x^2 + 4x - 3 = 0\)[/tex], we can use the quadratic formula, which is given by:
[tex]\[
x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}
\][/tex]
Here, the coefficients are [tex]\(A = 4\)[/tex], [tex]\(B = 4\)[/tex], and [tex]\(C = -3\)[/tex].
1. Calculate the discriminant:
The discriminant [tex]\(\Delta\)[/tex] is given by:
[tex]\[
\Delta = B^2 - 4AC
\][/tex]
Substitute the values of [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex]:
[tex]\[
\Delta = 4^2 - 4 \cdot 4 \cdot -3 = 16 + 48 = 64
\][/tex]
2. Calculate the two solutions:
Using the quadratic formula, we find the roots [tex]\(x_1\)[/tex] and [tex]\(x_2\)[/tex]:
[tex]\[
x_{1,2} = \frac{-B \pm \sqrt{\Delta}}{2A}
\][/tex]
Substitute the values of [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(\Delta\)[/tex]:
[tex]\[
x_{1,2} = \frac{-4 \pm \sqrt{64}}{2 \cdot 4} = \frac{-4 \pm 8}{8}
\][/tex]
3. Determine the individual roots:
- For [tex]\(x_1\)[/tex]:
[tex]\[
x_1 = \frac{-4 + 8}{8} = \frac{4}{8} = 0.5
\][/tex]
- For [tex]\(x_2\)[/tex]:
[tex]\[
x_2 = \frac{-4 - 8}{8} = \frac{-12}{8} = -1.5
\][/tex]
Therefore, the roots of the equation [tex]\(4x^2 + 4x - 3 = 0\)[/tex] are:
[tex]\[
x = 0.5 \quad \text{and} \quad x = -1.5
\][/tex]
The correct choice from the given options is:
[tex]\[
x = (-1.5, 0.5)
\][/tex]
