To solve the quadratic equation [tex]\( x^2 + 4x - 10 = 0 \)[/tex] using the quadratic formula, follow these steps:
1. Identify the coefficients: For the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex], the coefficients are:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 4 \)[/tex]
- [tex]\( c = -10 \)[/tex]
2. Calculate the discriminant: The discriminant ([tex]\( \Delta \)[/tex]) is given by the formula:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Plugging in the coefficients, we get:
[tex]\[
\Delta = 4^2 - 4 \cdot 1 \cdot (-10) = 16 + 40 = 56
\][/tex]
3. Evaluate the discriminant: Since the discriminant is positive ([tex]\( \Delta = 56 \)[/tex]), the quadratic equation has two distinct real solutions.
4. Use the quadratic formula: The quadratic formula to find the solutions is:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Substituting the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( \Delta \)[/tex], we get:
[tex]\[
x = \frac{-4 \pm \sqrt{56}}{2 \cdot 1} = \frac{-4 \pm \sqrt{56}}{2}
\][/tex]
5. Simplify the solutions:
- For the first solution ([tex]\( x_1 \)[/tex]):
[tex]\[
x_1 = \frac{-4 + \sqrt{56}}{2} \approx 1.7416573867739413
\][/tex]
- For the second solution ([tex]\( x_2 \)[/tex]):
[tex]\[
x_2 = \frac{-4 - \sqrt{56}}{2} \approx -5.741657386773941
\][/tex]
6. Write the solutions: The solutions are:
[tex]\[
x \approx 1.7416573867739413, \quad -5.741657386773941
\][/tex]
So the correct choice is:
A. [tex]\( x \approx 1.7416573867739413, -5.741657386773941 \)[/tex]
