To solve the quadratic equation [tex]\(5x^2 + 3x - 4 = 0\)[/tex] using the quadratic formula, we follow these steps:
1. Identify the coefficients: For the given quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex], the coefficients are:
- [tex]\(a = 5\)[/tex]
- [tex]\(b = 3\)[/tex]
- [tex]\(c = -4\)[/tex]
2. Quadratic Formula: The quadratic formula is given by:
[tex]\[
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}
\][/tex]
3. Substitute the coefficients into the formula:
- [tex]\(a = 5\)[/tex]
- [tex]\(b = 3\)[/tex]
- [tex]\(c = -4\)[/tex]
Substituting these values into the quadratic formula, we get:
[tex]\[
x = \frac{{-3 \pm \sqrt{{3^2 - 4 \cdot 5 \cdot (-4)}}}}{{2 \cdot 5}}
\][/tex]
4. Simplify inside the square root:
- Calculate the discriminant [tex]\(b^2 - 4ac\)[/tex]:
[tex]\[
3^2 - 4 \cdot 5 \cdot (-4) = 9 + 80 = 89
\][/tex]
5. Write out the final solution:
[tex]\[
x = \frac{{-3 \pm \sqrt{89}}}{{10}}
\][/tex]
Based on these steps, the correct form is:
[tex]\[
x = \frac{-3 \pm \sqrt{(3)^2 - 4(5)(-4)}}{2(5)}
\][/tex]
Thus, the correct equation that shows the quadratic formula used correctly is:
[tex]\[
x = \frac{-3 \pm \sqrt{(3)^2 - 4(5)(-4)}}{2(5)}
\][/tex]
