We need to solve four equations: 5) $(r+6)(r-6) = 0$ 6) $a(5a-4) = 0$ 7) $2(m-6)(8m-7) = 0$ 8) $3(7x-1)(8x-1) = 0$

1. Problem Description

We need to solve four equations:

5) $(r+6)(r-6) = 0$

6) $a(5a-4) = 0$

7) $2(m-6)(8m-7) = 0$

8) $3(7x-1)(8x-1) = 0$

2. Solution Steps

For each equation, we apply the zero-product property, which states that if the product of several factors is zero, then at least one of the factors must be zero.

5) $(r+6)(r-6) = 0$

Either $r+6 = 0$ or $r-6 = 0$.

If $r+6 = 0$, then $r = -6$.

If $r-6 = 0$, then $r = 6$.

6) $a(5a-4) = 0$

Either $a = 0$ or $5a-4 = 0$.

If $a = 0$, then $a = 0$.

If $5a-4 = 0$, then $5a = 4$, so $a = \frac{4}{5}$.

7) $2(m-6)(8m-7) = 0$

Either $2 = 0$, $m-6 = 0$, or $8m-7 = 0$.

Since $2 \neq 0$, we consider $m-6 = 0$ or $8m-7 = 0$.

If $m-6 = 0$, then $m = 6$.

If $8m-7 = 0$, then $8m = 7$, so $m = \frac{7}{8}$.

8) $3(7x-1)(8x-1) = 0$

Either $3 = 0$, $7x-1 = 0$, or $8x-1 = 0$.

Since $3 \neq 0$, we consider $7x-1 = 0$ or $8x-1 = 0$.

If $7x-1 = 0$, then $7x = 1$, so $x = \frac{1}{7}$.

If $8x-1 = 0$, then $8x = 1$, so $x = \frac{1}{8}$.

3. Final Answer

5) $r = -6$ or $r = 6$

6) $a = 0$ or $a = \frac{4}{5}$

7) $m = 6$ or $m = \frac{7}{8}$

8) $x = \frac{1}{7}$ or $x = \frac{1}{8}$