Let's solve the equation [tex]\(4x^2 + 4ax + \left(a^2 + b^2\right) = 0\)[/tex].
1. Identify the coefficients:
- The equation is of the form [tex]\(Ax^2 + Bx + C = 0\)[/tex]
- In this case:
[tex]\[A = 4, \quad B = 4a, \quad C = a^2 + b^2\][/tex]
2. Use the quadratic formula:
The quadratic formula states that the solutions of [tex]\(Ax^2 + Bx + C = 0\)[/tex] are given by:
[tex]\[
x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}
\][/tex]
3. Substitute the coefficients into the quadratic formula:
[tex]\[
x = \frac{-4a \pm \sqrt{(4a)^2 - 4 \cdot 4 \cdot (a^2 + b^2)}}{2 \cdot 4}
\][/tex]
4. Simplify inside the square root:
[tex]\[
x = \frac{-4a \pm \sqrt{16a^2 - 16(a^2 + b^2)}}{8}
\][/tex]
[tex]\[
x = \frac{-4a \pm \sqrt{16a^2 - 16a^2 - 16b^2}}{8}
\][/tex]
[tex]\[
x = \frac{-4a \pm \sqrt{-16b^2}}{8}
\][/tex]
5. Simplify the square root:
[tex]\[
x = \frac{-4a \pm 4i b}{8}
\][/tex]
6. Simplify the fractions:
[tex]\[
x = \frac{-4a}{8} \pm \frac{4ib}{8}
\][/tex]
[tex]\[
x = \frac{-a}{2} \pm \frac{ib}{2}
\][/tex]
7. Express the solutions:
[tex]\[
x = -\frac{a}{2} - \frac{ib}{2} \quad \text{and} \quad x = -\frac{a}{2} + \frac{ib}{2}
\][/tex]
Therefore, the solutions to the quadratic equation [tex]\(4x^2 + 4ax + (a^2 + b^2) = 0\)[/tex] are:
[tex]\[
x = -\frac{a}{2} - \frac{ib}{2} \quad \text{and} \quad x = -\frac{a}{2} + \frac{ib}{2}
\][/tex]
