Question

type the correct answer in the box.
solve the following quadratic equation using the quadratic formula.
$5x^2 - 8x + 5 = 0$
write the solutions in the following form, where $r$, $s$, and $t$ are integers, and the fractions are in simplest form.
$x = \frac{r - s\mathrm{i}}{t}, x = \frac{r + s\mathrm{i}}{t}$

Explanation:

Step1: Identify quadratic coefficients

For $5x^2 - 8x + 5 = 0$, $a=5$, $b=-8$, $c=5$

Step2: Recall quadratic formula

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Step3: Compute discriminant

$\sqrt{(-8)^2 - 4(5)(5)} = \sqrt{64 - 100} = \sqrt{-36} = 6i$

Step4: Substitute into formula

$x = \frac{8 \pm 6i}{2(5)} = \frac{8 \pm 6i}{10}$

Step5: Simplify the fraction

$\frac{8 \pm 6i}{10} = \frac{4 \pm 3i}{5}$

Answer:

$x = \frac{4 - 3i}{5}, x = \frac{4 + 3i}{5}$