Solve the following quadratic and linear equations: 5.1 2x2 = -x 5.2 x2 − 25 = 0 5.3 x2 + 5x − 6 = 0 5.4 racx+53 + racx+16 = racx3 - 2

$$x = 0 \quad \text{or} \quad 2x + 1 = 0$$ $$2x = -1$$ $$x = -\frac{1}{2}$$

The solutions are $\boxed{x = 0 \text{ or } x = -\frac{1}{2}}$.

Remember to include both positive and negative roots.

$$\sqrt{x^2} = \pm\sqrt{25}$$ $$x = \pm 5$$

The solutions are $\boxed{x = 5 \text{ or } x = -5}$.

Find two numbers that multiply to $-6$ and add to 5. These numbers are 6 and $-1$.

$$(x + 6)(x - 1) = 0$$

$$x + 6 = 0 \quad \text{or} \quad x - 1 = 0$$ $$x = -6 \quad \text{or} \quad x = 1$$

The solutions are $\boxed{x = -6 \text{ or } x = 1}$.

Multiply every term by the LCD to clear the denominators.

$$6 \left(\frac{x+5}{3}\right) + 6 \left(\frac{x+1}{6}\right) = 6 \left(\frac{x}{3}\right) - 6(2)$$ $$2(x+5) + (x+1) = 2x - 12$$

$$2x + 10 + x + 1 = 2x - 12$$ $$3x + 11 = 2x - 12$$

$$3x - 2x = -12 - 11$$ $$x = -23$$

The solution is $\boxed{x = -23}$.

Let $f$ be the number of 50c coins.
Let $t$ be the number of 20c coins.
Total number of coins:

$$f + t = 120 \quad (1)$$

Total value (in Rands):

$$0.50f + 0.20t = 34.80 \quad (2)$$

$$0.50f + 24 - 0.20f = 34.80$$ $$0.30f + 24 = 34.80$$ $$0.30f = 34.80 - 24$$ $$0.30f = 10.80$$ $$f = \frac{10.80}{0.30}$$ $$f = 36$$

There are $\boxed{36}$ 50c coins.