To solve the quadratic equation [tex]\(2x^2 + 10x - 12 = 0\)[/tex], we can proceed with the following steps:
1. Rewrite the equation in standard quadratic form:
[tex]\(2x^2 + 10x - 12 = 0\)[/tex]
2. Identify the coefficients:
Here, [tex]\(a = 2\)[/tex], [tex]\(b = -10\)[/tex], and [tex]\(c = 12\)[/tex].
3. Calculate the discriminant:
The discriminant [tex]\(\Delta\)[/tex] can be found using the formula
[tex]\(\Delta = b^2 - 4ac\)[/tex].
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the formula:
[tex]\[
\Delta = (-10)^2 - 4 \cdot 2 \cdot 12
\][/tex]
[tex]\[
\Delta = 100 - 96
\][/tex]
[tex]\[
\Delta = 4
\][/tex]
4. Check the nature of the roots:
Since the discriminant is positive ([tex]\(\Delta > 0\)[/tex]), we will have two distinct real roots.
5. Calculate the roots using the quadratic formula:
The quadratic formula is given by:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(\Delta\)[/tex]:
[tex]\[
x = \frac{-(-10) \pm \sqrt{4}}{2 \cdot 2}
\][/tex]
[tex]\[
x = \frac{10 \pm 2}{4}
\][/tex]
6. Find the individual roots:
[tex]\[
x_1 = \frac{10 + 2}{4} = \frac{12}{4} = 3
\][/tex]
[tex]\[
x_2 = \frac{10 - 2}{4} = \frac{8}{4} = 2
\][/tex]
7. Check if the solutions match the provided options:
Compare the solutions [tex]\((3, 2)\)[/tex] with the given options [tex]\(x = 6, x = -2, x = 3, x = 1, x = -6, x = -3\)[/tex]:
The only matching solution is [tex]\(x = 3\)[/tex].
Thus, the correct solution to the equation [tex]\(2x^2 + 10x - 12 = 0\)[/tex] that is among the provided options is:
[tex]\[ x = 3 \][/tex]
Hence, the answer is:
[tex]\[ x = 3 \][/tex]
