Sure! Let's solve the quadratic equation step by step:
Given the quadratic equation:
[tex]\[8x^2 + 2x - 15 = 0\][/tex]
We will use the quadratic formula to find the roots of this equation. The quadratic formula is:
[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]. In this equation:
[tex]\[a = 8\][/tex]
[tex]\[b = 2\][/tex]
[tex]\[c = -15\][/tex]
### Step-by-Step Solution:
1. Calculate the Discriminant:
The discriminant ([tex]\(\Delta\)[/tex]) of a quadratic equation is given by the expression:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting the values [tex]\(a = 8\)[/tex], [tex]\(b = 2\)[/tex], and [tex]\(c = -15\)[/tex]:
[tex]\[
\Delta = 2^2 - 4 \cdot 8 \cdot (-15)
\][/tex]
Simplifying this expression:
[tex]\[
\Delta = 4 + 480 = 484
\][/tex]
2. Calculate the Square Root of the Discriminant:
Next, we find the square root of the discriminant:
[tex]\[
\sqrt{\Delta} = \sqrt{484} = 22
\][/tex]
3. Find the Two Solutions:
Using the quadratic formula, we have two solutions:
[tex]\[
x_1 = \frac{-b + \sqrt{\Delta}}{2a}
\][/tex]
[tex]\[
x_2 = \frac{-b - \sqrt{\Delta}}{2a}
\][/tex]
Now substitute [tex]\( b = 2 \)[/tex], [tex]\( \sqrt{\Delta} = 22 \)[/tex], and [tex]\( a = 8 \)[/tex]:
First root [tex]\( x_1 \)[/tex]:
[tex]\[
x_1 = \frac{-2 + 22}{2 \cdot 8} = \frac{20}{16} = 1.25
\][/tex]
Second root [tex]\( x_2 \)[/tex]:
[tex]\[
x_2 = \frac{-2 - 22}{2 \cdot 8} = \frac{-24}{16} = -1.5
\][/tex]
So, the solutions to the quadratic equation [tex]\(8x^2 + 2x - 15 = 0\)[/tex] are:
[tex]\[ x = 1.25 \text{ and } x = -1.5 \][/tex]
To summarize:
- Discriminant ([tex]\(\Delta\)[/tex]): [tex]\(484\)[/tex]
- Square root of the discriminant: [tex]\(22\)[/tex]
- Roots of the quadratic equation: [tex]\(x_1 = 1.25\)[/tex] and [tex]\(x_2 = -1.5\)[/tex]
These are the detailed steps to solve the given quadratic equation.
