Solve 0=-5250+75t^2-225t

$$0 = - 5250 + 75 t ^ { 2 } - 225 t$$

Solve for t

$$t=-7$$

$$t=10$$

Steps Using Factoring By Grouping

Swap sides so that all variable terms are on the left hand side.

$$-5250+75t^{2}-225t=0$$

Divide both sides by 75.

$$-70+t^{2}-3t=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$t^{2}-3t-70=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as t^{2}+at+bt-70. To find a and b, set up a system to be solved.

$$a+b=-3 ab=1\left(-70\right)=-70$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -70.

$$1,-70 2,-35 5,-14 7,-10$$

Calculate the sum for each pair.

$$1-70=-69 2-35=-33 5-14=-9 7-10=-3$$

The solution is the pair that gives sum -3.

$$a=-10 b=7$$

Rewrite t^{2}-3t-70 as \left(t^{2}-10t\right)+\left(7t-70\right).

$$\left(t^{2}-10t\right)+\left(7t-70\right)$$

Factor out t in the first and 7 in the second group.

$$t\left(t-10\right)+7\left(t-10\right)$$

Factor out common term t-10 by using distributive property.

$$\left(t-10\right)\left(t+7\right)$$

To find equation solutions, solve t-10=0 and t+7=0.

$$t=10 t=-7$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$-5250+75t^{2}-225t=0$$

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.

$$75t^{2}-225t-5250=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 75 for a, -225 for b, and -5250 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-225\right)±\sqrt{\left(-225\right)^{2}-4\times 75\left(-5250\right)}}{2\times 75}$$

Square -225.

$$t=\frac{-\left(-225\right)±\sqrt{50625-4\times 75\left(-5250\right)}}{2\times 75}$$

Multiply -4 times 75.

$$t=\frac{-\left(-225\right)±\sqrt{50625-300\left(-5250\right)}}{2\times 75}$$

Multiply -300 times -5250.

$$t=\frac{-\left(-225\right)±\sqrt{50625+1575000}}{2\times 75}$$

Add 50625 to 1575000.

$$t=\frac{-\left(-225\right)±\sqrt{1625625}}{2\times 75}$$

Take the square root of 1625625.

$$t=\frac{-\left(-225\right)±1275}{2\times 75}$$

The opposite of -225 is 225.

$$t=\frac{225±1275}{2\times 75}$$

Multiply 2 times 75.

$$t=\frac{225±1275}{150}$$

Now solve the equation t=\frac{225±1275}{150} when ± is plus. Add 225 to 1275.

$$t=\frac{1500}{150}$$

Divide 1500 by 150.

$$t=10$$

Now solve the equation t=\frac{225±1275}{150} when ± is minus. Subtract 1275 from 225.

$$t=-\frac{1050}{150}$$

Divide -1050 by 150.

$$t=-7$$

The equation is now solved.

$$t=10 t=-7$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$-5250+75t^{2}-225t=0$$

Add 5250 to both sides. Anything plus zero gives itself.

$$75t^{2}-225t=5250$$

Divide both sides by 75.

$$\frac{75t^{2}-225t}{75}=\frac{5250}{75}$$

Dividing by 75 undoes the multiplication by 75.

$$t^{2}+\left(-\frac{225}{75}\right)t=\frac{5250}{75}$$

Divide -225 by 75.

$$t^{2}-3t=\frac{5250}{75}$$

Divide 5250 by 75.

$$t^{2}-3t=70$$

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

$$t^{2}-3t+\left(-\frac{3}{2}\right)^{2}=70+\left(-\frac{3}{2}\right)^{2}$$

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-3t+\frac{9}{4}=70+\frac{9}{4}$$

Add 70 to \frac{9}{4}.

$$t^{2}-3t+\frac{9}{4}=\frac{289}{4}$$

Factor t^{2}-3t+\frac{9}{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}.

$$\left(t-\frac{3}{2}\right)^{2}=\frac{289}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{3}{2}\right)^{2}}=\sqrt{\frac{289}{4}}$$

Simplify.

$$t-\frac{3}{2}=\frac{17}{2} t-\frac{3}{2}=-\frac{17}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$t=10 t=-7$$