Solve (t+2)(1/2t-4)=25/4

$$( t + 2 ) ( \frac { 1 } { 2 } t - 4 ) = \frac { 25 } { 4 }$$

Solve for t

$$t=\frac{5\sqrt{6}}{2}+3\approx 9.123724357$$

$$t=-\frac{5\sqrt{6}}{2}+3\approx -3.123724357$$

Steps Using the Quadratic Formula

Use the distributive property to multiply t+2 by \frac{1}{2}t-4 and combine like terms.

$$\frac{1}{2}t^{2}-3t-8=\frac{25}{4}$$

Subtract \frac{25}{4} from both sides.

$$\frac{1}{2}t^{2}-3t-8-\frac{25}{4}=0$$

Subtract \frac{25}{4} from -8 to get -\frac{57}{4}.

$$\frac{1}{2}t^{2}-3t-\frac{57}{4}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{2} for a, -3 for b, and -\frac{57}{4} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times \frac{1}{2}\left(-\frac{57}{4}\right)}}{2\times \frac{1}{2}}$$

Square -3.

$$t=\frac{-\left(-3\right)±\sqrt{9-4\times \frac{1}{2}\left(-\frac{57}{4}\right)}}{2\times \frac{1}{2}}$$

Multiply -4 times \frac{1}{2}.

$$t=\frac{-\left(-3\right)±\sqrt{9-2\left(-\frac{57}{4}\right)}}{2\times \frac{1}{2}}$$

Multiply -2 times -\frac{57}{4}.

$$t=\frac{-\left(-3\right)±\sqrt{9+\frac{57}{2}}}{2\times \frac{1}{2}}$$

Add 9 to \frac{57}{2}.

$$t=\frac{-\left(-3\right)±\sqrt{\frac{75}{2}}}{2\times \frac{1}{2}}$$

Take the square root of \frac{75}{2}.

$$t=\frac{-\left(-3\right)±\frac{5\sqrt{6}}{2}}{2\times \frac{1}{2}}$$

The opposite of -3 is 3.

$$t=\frac{3±\frac{5\sqrt{6}}{2}}{2\times \frac{1}{2}}$$

Multiply 2 times \frac{1}{2}.

$$t=\frac{3±\frac{5\sqrt{6}}{2}}{1}$$

Now solve the equation t=\frac{3±\frac{5\sqrt{6}}{2}}{1} when ± is plus. Add 3 to \frac{5\sqrt{6}}{2}.

$$t=\frac{\frac{5\sqrt{6}}{2}+3}{1}$$

Divide 3+\frac{5\sqrt{6}}{2} by 1.

$$t=\frac{5\sqrt{6}}{2}+3$$

Now solve the equation t=\frac{3±\frac{5\sqrt{6}}{2}}{1} when ± is minus. Subtract \frac{5\sqrt{6}}{2} from 3.

$$t=\frac{-\frac{5\sqrt{6}}{2}+3}{1}$$

Divide 3-\frac{5\sqrt{6}}{2} by 1.

$$t=-\frac{5\sqrt{6}}{2}+3$$

The equation is now solved.

$$t=\frac{5\sqrt{6}}{2}+3 t=-\frac{5\sqrt{6}}{2}+3$$

Steps for Completing the Square

Use the distributive property to multiply t+2 by \frac{1}{2}t-4 and combine like terms.

$$\frac{1}{2}t^{2}-3t-8=\frac{25}{4}$$

Add 8 to both sides.

$$\frac{1}{2}t^{2}-3t=\frac{25}{4}+8$$

Add \frac{25}{4} and 8 to get \frac{57}{4}.

$$\frac{1}{2}t^{2}-3t=\frac{57}{4}$$

Multiply both sides by 2.

$$\frac{\frac{1}{2}t^{2}-3t}{\frac{1}{2}}=\frac{\frac{57}{4}}{\frac{1}{2}}$$

Dividing by \frac{1}{2} undoes the multiplication by \frac{1}{2}.

$$t^{2}+\left(-\frac{3}{\frac{1}{2}}\right)t=\frac{\frac{57}{4}}{\frac{1}{2}}$$

Divide -3 by \frac{1}{2} by multiplying -3 by the reciprocal of \frac{1}{2}.

$$t^{2}-6t=\frac{\frac{57}{4}}{\frac{1}{2}}$$

Divide \frac{57}{4} by \frac{1}{2} by multiplying \frac{57}{4} by the reciprocal of \frac{1}{2}.

$$t^{2}-6t=\frac{57}{2}$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-6t+\left(-3\right)^{2}=\frac{57}{2}+\left(-3\right)^{2}$$

Square -3.

$$t^{2}-6t+9=\frac{57}{2}+9$$

Add \frac{57}{2} to 9.

$$t^{2}-6t+9=\frac{75}{2}$$

Factor t^{2}-6t+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-3\right)^{2}=\frac{75}{2}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-3\right)^{2}}=\sqrt{\frac{75}{2}}$$

Simplify.

$$t-3=\frac{5\sqrt{6}}{2} t-3=-\frac{5\sqrt{6}}{2}$$

Add 3 to both sides of the equation.

$$t=\frac{5\sqrt{6}}{2}+3 t=-\frac{5\sqrt{6}}{2}+3$$