# How do you solve the equation 4+sqrt(4t-4)=t?

Oct 17, 2016

$t = 2$ or $t = 10$

#### Explanation:

$4 + \sqrt{4 t - 4} = t$

Subtract 4 from both sides.

$\sqrt{4 t - 4} = t - 4$

Square both sides.

$4 t - 4 = {\left(t - 4\right)}^{2}$

Expand the right side.

$4 t - 4 = \left(t - 4\right) \left(t - 4\right)$

$4 t - 4 = t \left(t - 4\right) - 4 \left(t - 4\right)$

$4 t - 4 = {t}^{2} - 4 t - 4 t + 16$ (since multiplying two negatives produces a positive)

Simplify the equation on the right.

$4 t - 4 = {t}^{2} - 8 t + 16$

Subtract $\left(4 t - 4\right)$ from both sides.

$0 = {t}^{2} - 12 t + 20$ or ${t}^{2} - 12 t + 20 = 0$

Factorise the equation.

${t}^{2} - 2 t - 10 t + 20 = 0$

$t \left(t - 2\right) - 10 \left(t - 2\right) = 0$

$\left(t - 2\right) \left(t - 10\right) = 0$

$\therefore t = 2$ or $t = 10$