Certainly! Let's solve the quadratic equation [tex]\(2x^2 + 7x - 4 = 0\)[/tex] step by step.
1. Identify the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
The quadratic equation is given in standard form [tex]\(ax^2 + bx + c = 0\)[/tex].
- [tex]\(a = 2\)[/tex]
- [tex]\(b = 7\)[/tex]
- [tex]\(c = -4\)[/tex]
2. Write down the quadratic formula:
The quadratic formula for finding the roots of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
3. Calculate the discriminant:
The discriminant is the part of the quadratic formula under the square root:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting in the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
\Delta = 7^2 - 4(2)(-4) = 49 + 32 = 81
\][/tex]
4. Calculate the two roots:
Using the discriminant, we now find the two possible values for [tex]\(x\)[/tex] using the plus and minus in the quadratic formula.
- First root:
[tex]\[
x_1 = \frac{-b + \sqrt{\Delta}}{2a} = \frac{-7 + \sqrt{81}}{2 \cdot 2} = \frac{-7 + 9}{4} = \frac{2}{4} = 0.5
\][/tex]
- Second root:
[tex]\[
x_2 = \frac{-b - \sqrt{\Delta}}{2a} = \frac{-7 - \sqrt{81}}{2 \cdot 2} = \frac{-7 - 9}{4} = \frac{-16}{4} = -4
\][/tex]
5. Summary:
- The discriminant [tex]\(\Delta\)[/tex] is 81.
- The first root [tex]\(x_1\)[/tex] is 0.5.
- The second root [tex]\(x_2\)[/tex] is -4.0.
So the solutions of the quadratic equation [tex]\(2x^2 + 7x - 4 = 0\)[/tex] are:
[tex]\[
x_1 = 0.5 \quad \text{and} \quad x_2 = -4
\][/tex]
