To solve the quadratic equation [tex]\(2x^2 - 5x - 7 = 0\)[/tex], we can use the quadratic formula, which states:
[tex]\[
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}
\][/tex]
where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are the coefficients of the equation [tex]\(ax^2 + bx + c = 0\)[/tex].
1. Identify the coefficients:
[tex]\[
a = 2, \quad b = -5, \quad c = -7
\][/tex]
2. Calculate the discriminant:
The discriminant [tex]\(\Delta\)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
\Delta = (-5)^2 - 4 \cdot 2 \cdot (-7)
\][/tex]
[tex]\[
\Delta = 25 + 56
\][/tex]
[tex]\[
\Delta = 81
\][/tex]
3. Find the roots using the quadratic formula:
Since the discriminant is positive ([tex]\(\Delta = 81\)[/tex]), we have two distinct real roots. The roots are calculated as:
[tex]\[
x_1 = \frac{{-(-5) + \sqrt{81}}}{2 \cdot 2} \quad \text{and} \quad x_2 = \frac{{-(-5) - \sqrt{81}}}{2 \cdot 2}
\][/tex]
Simplifying each case:
- For [tex]\(x_1\)[/tex]:
[tex]\[
x_1 = \frac{{5 + 9}}{4} = \frac{{14}}{4} = 3.5
\][/tex]
- For [tex]\(x_2\)[/tex]:
[tex]\[
x_2 = \frac{{5 - 9}}{4} = \frac{{-4}}{4} = -1.0
\][/tex]
Thus, the two solutions for the equation [tex]\(2x^2 - 5x - 7 = 0\)[/tex] are:
[tex]\[
x_1 = 3.5 \quad \text{and} \quad x_2 = -1.0
\][/tex]
Therefore, the possible values of [tex]\(x\)[/tex] are:
[tex]\[
\boxed{3.5 \quad \text{and} \quad -1.0}
\][/tex]
