To solve the quadratic equation [tex]\(8x^2 - 5x - 21 = 0\)[/tex], we need to use the quadratic formula, which is given by:
[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 from the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex].
For the given equation [tex]\(8x^2 - 5x - 21 = 0\)[/tex]:
- [tex]\(a = 8\)[/tex]
- [tex]\(b = -5\)[/tex]
- [tex]\(c = -21\)[/tex]
Step 1: Calculate the discriminant
The discriminant ([tex]\(\Delta\)[/tex]) is given by [tex]\(b^2 - 4ac\)[/tex]:
[tex]\[
\Delta = (-5)^2 - 4 \cdot 8 \cdot (-21)
\][/tex]
[tex]\[
\Delta = 25 + 672
\][/tex]
[tex]\[
\Delta = 697
\][/tex]
Step 2: Calculate the solutions using the quadratic formula
Since the discriminant is positive ([tex]\(\Delta = 697\)[/tex]), there are two real solutions given by:
[tex]\[
x_1 = \frac{-b + \sqrt{\Delta}}{2a}
\][/tex]
[tex]\[
x_2 = \frac{-b - \sqrt{\Delta}}{2a}
\][/tex]
Plugging in the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(\Delta\)[/tex]:
[tex]\[
x_1 = \frac{-(-5) + \sqrt{697}}{2 \cdot 8}
\][/tex]
[tex]\[
x_1 = \frac{5 + \sqrt{697}}{16}
\][/tex]
[tex]\[
x_1 \approx 1.9625
\][/tex]
And for the second solution:
[tex]\[
x_2 = \frac{-(-5) - \sqrt{697}}{2 \cdot 8}
\][/tex]
[tex]\[
x_2 = \frac{5 - \sqrt{697}}{16}
\][/tex]
[tex]\[
x_2 \approx -1.3375
\][/tex]
Conclusion:
The solutions to the quadratic equation [tex]\(8x^2 - 5x - 21 = 0\)[/tex] are approximately:
[tex]\[
x_1 \approx 1.9625
\][/tex]
[tex]\[
x_2 \approx -1.3375
\][/tex]
The discriminant value is [tex]\(697\)[/tex], confirming that there are two distinct real roots for the equation.
