Solve 5t-15=15-5t/5t

$$5 t - 15 = 15 - \frac { 5 t } { 5 } t$$

Solve for t

$$t = \frac{\sqrt{145} - 5}{2} \approx 3.520797289$$

$$t=\frac{-\sqrt{145}-5}{2}\approx -8.520797289$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 5.

$$25t-75=75-5tt$$

Multiply t and t to get t^{2}.

$$25t-75=75-5t^{2}$$

Subtract 75 from both sides.

$$25t-75-75=-5t^{2}$$

Subtract 75 from -75 to get -150.

$$25t-150=-5t^{2}$$

Add 5t^{2} to both sides.

$$25t-150+5t^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$5t^{2}+25t-150=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 25 for b, and -150 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-25±\sqrt{25^{2}-4\times 5\left(-150\right)}}{2\times 5}$$

Square 25.

$$t=\frac{-25±\sqrt{625-4\times 5\left(-150\right)}}{2\times 5}$$

Multiply -4 times 5.

$$t=\frac{-25±\sqrt{625-20\left(-150\right)}}{2\times 5}$$

Multiply -20 times -150.

$$t=\frac{-25±\sqrt{625+3000}}{2\times 5}$$

Add 625 to 3000.

$$t=\frac{-25±\sqrt{3625}}{2\times 5}$$

Take the square root of 3625.

$$t=\frac{-25±5\sqrt{145}}{2\times 5}$$

Multiply 2 times 5.

$$t=\frac{-25±5\sqrt{145}}{10}$$

Now solve the equation t=\frac{-25±5\sqrt{145}}{10} when ± is plus. Add -25 to 5\sqrt{145}.

$$t=\frac{5\sqrt{145}-25}{10}$$

Divide -25+5\sqrt{145} by 10.

$$t=\frac{\sqrt{145}-5}{2}$$

Now solve the equation t=\frac{-25±5\sqrt{145}}{10} when ± is minus. Subtract 5\sqrt{145} from -25.

$$t=\frac{-5\sqrt{145}-25}{10}$$

Divide -25-5\sqrt{145} by 10.

$$t=\frac{-\sqrt{145}-5}{2}$$

The equation is now solved.

$$t=\frac{\sqrt{145}-5}{2} t=\frac{-\sqrt{145}-5}{2}$$

Steps for Completing the Square

Multiply both sides of the equation by 5.

$$25t-75=75-5tt$$

Multiply t and t to get t^{2}.

$$25t-75=75-5t^{2}$$

Add 5t^{2} to both sides.

$$25t-75+5t^{2}=75$$

Add 75 to both sides.

$$25t+5t^{2}=75+75$$

Add 75 and 75 to get 150.

$$25t+5t^{2}=150$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$5t^{2}+25t=150$$

Divide both sides by 5.

$$\frac{5t^{2}+25t}{5}=\frac{150}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$t^{2}+\frac{25}{5}t=\frac{150}{5}$$

Divide 25 by 5.

$$t^{2}+5t=\frac{150}{5}$$

Divide 150 by 5.

$$t^{2}+5t=30$$

Divide 5, the coefficient of the x term, by 2 to get \frac{5}{2}. Then add the square of \frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}+5t+\left(\frac{5}{2}\right)^{2}=30+\left(\frac{5}{2}\right)^{2}$$

Square \frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$t^{2}+5t+\frac{25}{4}=30+\frac{25}{4}$$

Add 30 to \frac{25}{4}.

$$t^{2}+5t+\frac{25}{4}=\frac{145}{4}$$

Factor t^{2}+5t+\frac{25}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t+\frac{5}{2}\right)^{2}=\frac{145}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t+\frac{5}{2}\right)^{2}}=\sqrt{\frac{145}{4}}$$

Simplify.

$$t+\frac{5}{2}=\frac{\sqrt{145}}{2} t+\frac{5}{2}=-\frac{\sqrt{145}}{2}$$

Subtract \frac{5}{2} from both sides of the equation.

$$t=\frac{\sqrt{145}-5}{2} t=\frac{-\sqrt{145}-5}{2}$$