We are given two quadratic equations: e) $5x^2 - 5x - 3 = 0$ f) $x^2 = -2x + 1$ We need to solve each equation for $x$.

1. Problem Description

We are given two quadratic equations:

e) $5x^2 - 5x - 3 = 0$

f) $x^2 = -2x + 1$

We need to solve each equation for $x$.

2. Solution Steps

e) We have the quadratic equation $5x^2 - 5x - 3 = 0$. We will use the quadratic formula to find the solutions.

The quadratic formula is given by:

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

In this case, $a = 5$, $b = -5$, and $c = -3$.

Plugging these values into the quadratic formula, we get:

$x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(5)(-3)}}{2(5)}$

$x = \frac{5 \pm \sqrt{25 + 60}}{10}$

$x = \frac{5 \pm \sqrt{85}}{10}$

Thus, the solutions are $x = \frac{5 + \sqrt{85}}{10}$ and $x = \frac{5 - \sqrt{85}}{10}$.

f) We have the equation $x^2 = -2x + 1$. First, we rewrite the equation in standard quadratic form:

$x^2 + 2x - 1 = 0$

Now, we can use the quadratic formula with $a = 1$, $b = 2$, and $c = -1$:

$x = \frac{-2 \pm \sqrt{2^2 - 4(1)(-1)}}{2(1)}$

$x = \frac{-2 \pm \sqrt{4 + 4}}{2}$

$x = \frac{-2 \pm \sqrt{8}}{2}$

$x = \frac{-2 \pm 2\sqrt{2}}{2}$

$x = -1 \pm \sqrt{2}$

Thus, the solutions are $x = -1 + \sqrt{2}$ and $x = -1 - \sqrt{2}$.

3. Final Answer

e) $x = \frac{5 + \sqrt{85}}{10}$, $x = \frac{5 - \sqrt{85}}{10}$

f) $x = -1 + \sqrt{2}$, $x = -1 - \sqrt{2}$