To solve the quadratic equation [tex]\(0 = 2x^2 - 10x + 7\)[/tex] using the quadratic formula, follow these steps:
1. Identify the coefficients:
The given quadratic equation is in the standard form [tex]\(ax^2 + bx + c = 0\)[/tex].
Here, [tex]\(a = 2\)[/tex], [tex]\(b = -10\)[/tex], and [tex]\(c = 7\)[/tex].
2. Calculate the discriminant:
The discriminant [tex]\(\Delta\)[/tex] of the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by the formula:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
\Delta = (-10)^2 - 4(2)(7) = 100 - 56 = 44
\][/tex]
3. Apply the quadratic formula:
The quadratic formula to find the roots of the equation [tex]\(ax^2 + bx + c = 0\)[/tex] is:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
4. Compute the two solutions:
Using the discriminant [tex]\(\Delta = 44\)[/tex], substitute the values into the quadratic formula.
For the first solution:
[tex]\[
x_1 = \frac{-(-10) + \sqrt{44}}{2(2)} = \frac{10 + \sqrt{44}}{4}
\][/tex]
Evaluating this expression gives:
[tex]\[
x_1 \approx 4.16 \text{ (rounded to 2 decimal places)}
\][/tex]
For the second solution:
[tex]\[
x_2 = \frac{-(-10) - \sqrt{44}}{2(2)} = \frac{10 - \sqrt{44}}{4}
\][/tex]
Evaluating this expression gives:
[tex]\[
x_2 \approx 0.84 \text{ (rounded to 2 decimal places)}
\][/tex]
So, the two solutions to the equation [tex]\(0 = 2x^2 - 10x + 7\)[/tex] are approximately:
[tex]\[
x_1 = 4.16 \quad \text{and} \quad x_2 = 0.84
\][/tex]
