Solve (t-6)(t-8)=24

$$( t - 6 ) ( t - 8 ) = 24$$

Solve for t

$$t=2$$

$$t=12$$

Steps Using the Quadratic Formula

Use the distributive property to multiply t-6 by t-8 and combine like terms.

$$t^{2}-14t+48=24$$

Subtract 24 from both sides.

$$t^{2}-14t+48-24=0$$

Subtract 24 from 48 to get 24.

$$t^{2}-14t+24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -14 for b, and 24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 24}}{2}$$

Square -14.

$$t=\frac{-\left(-14\right)±\sqrt{196-4\times 24}}{2}$$

Multiply -4 times 24.

$$t=\frac{-\left(-14\right)±\sqrt{196-96}}{2}$$

Add 196 to -96.

$$t=\frac{-\left(-14\right)±\sqrt{100}}{2}$$

Take the square root of 100.

$$t=\frac{-\left(-14\right)±10}{2}$$

The opposite of -14 is 14.

$$t=\frac{14±10}{2}$$

Now solve the equation t=\frac{14±10}{2} when ± is plus. Add 14 to 10.

$$t=\frac{24}{2}$$

Divide 24 by 2.

$$t=12$$

Now solve the equation t=\frac{14±10}{2} when ± is minus. Subtract 10 from 14.

$$t=\frac{4}{2}$$

Divide 4 by 2.

$$t=2$$

The equation is now solved.

$$t=12 t=2$$

Steps for Completing the Square

Use the distributive property to multiply t-6 by t-8 and combine like terms.

$$t^{2}-14t+48=24$$

Subtract 48 from both sides.

$$t^{2}-14t=24-48$$

Subtract 48 from 24 to get -24.

$$t^{2}-14t=-24$$

Divide -14, the coefficient of the x term, by 2 to get -7. Then add the square of -7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-14t+\left(-7\right)^{2}=-24+\left(-7\right)^{2}$$

Square -7.

$$t^{2}-14t+49=-24+49$$

Add -24 to 49.

$$t^{2}-14t+49=25$$

Factor t^{2}-14t+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-7\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-7\right)^{2}}=\sqrt{25}$$

Simplify.

$$t-7=5 t-7=-5$$

Add 7 to both sides of the equation.

$$t=12 t=2$$