Certainly! Let's solve the quadratic equation [tex]\(x^2 + 10x + 2 = 0\)[/tex] step by step.
1. Start with the given quadratic equation:
[tex]\[
x^2 + 10x + 2 = 0
\][/tex]
2. Move the constant term to the right side of the equation:
[tex]\[
x^2 + 10x = -2
\][/tex]
3. To complete the square, take the coefficient of [tex]\(x\)[/tex], which is 10, divide it by 2, and square the result. This gives:
[tex]\[
\left(\frac{10}{2}\right)^2 = 25
\][/tex]
4. Add this value (25) to both sides of the equation:
[tex]\[
x^2 + 10x + 25 = -2 + 25
\][/tex]
5. Simplify the right side:
[tex]\[
x^2 + 10x + 25 = 23
\][/tex]
6. Now, the left side of the equation is a perfect square:
[tex]\[
(x + 5)^2 = 23
\][/tex]
7. Take the square root of both sides of the equation:
[tex]\[
x + 5 = \pm \sqrt{23}
\][/tex]
8. Solve for [tex]\(x\)[/tex] by isolating it on one side:
[tex]\[
x = -5 + \sqrt{23} \quad \text{or} \quad x = -5 - \sqrt{23}
\][/tex]
Therefore, the solutions to the quadratic equation [tex]\(x^2 + 10x + 2 = 0\)[/tex] are:
[tex]\[
x = -5 + \sqrt{23} \approx 0.196 \quad \text{and} \quad x = -5 - \sqrt{23} \approx -10.196
\][/tex]
So, the solutions are roughly [tex]\(0.196\)[/tex] and [tex]\(-10.196\)[/tex].
