Solve -t^2+20t+175=0

$$- { t }^{ 2 } +20t+175 = 0$$

Solve for t

$$t=5\sqrt{11}+10\approx 26.583123952$$

$$t=10-5\sqrt{11}\approx -6.583123952$$

Steps Using the Quadratic Formula

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.

$$-t^{2}+20t+175=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 20 for b, and 175 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-20±\sqrt{20^{2}-4\left(-1\right)\times 175}}{2\left(-1\right)}$$

Square 20.

$$t=\frac{-20±\sqrt{400-4\left(-1\right)\times 175}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$t=\frac{-20±\sqrt{400+4\times 175}}{2\left(-1\right)}$$

Multiply 4 times 175.

$$t=\frac{-20±\sqrt{400+700}}{2\left(-1\right)}$$

Add 400 to 700.

$$t=\frac{-20±\sqrt{1100}}{2\left(-1\right)}$$

Take the square root of 1100.

$$t=\frac{-20±10\sqrt{11}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$t=\frac{-20±10\sqrt{11}}{-2}$$

Now solve the equation t=\frac{-20±10\sqrt{11}}{-2} when ± is plus. Add -20 to 10\sqrt{11}.

$$t=\frac{10\sqrt{11}-20}{-2}$$

Divide -20+10\sqrt{11} by -2.

$$t=10-5\sqrt{11}$$

Now solve the equation t=\frac{-20±10\sqrt{11}}{-2} when ± is minus. Subtract 10\sqrt{11} from -20.

$$t=\frac{-10\sqrt{11}-20}{-2}$$

Divide -20-10\sqrt{11} by -2.

$$t=5\sqrt{11}+10$$

The equation is now solved.

$$t=10-5\sqrt{11} t=5\sqrt{11}+10$$

Steps for Completing the Square

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.

$$-t^{2}+20t+175=0$$

Subtract 175 from both sides of the equation.

$$-t^{2}+20t+175-175=-175$$

Subtracting 175 from itself leaves 0.

$$-t^{2}+20t=-175$$

Divide both sides by -1.

$$\frac{-t^{2}+20t}{-1}=-\frac{175}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$t^{2}+\frac{20}{-1}t=-\frac{175}{-1}$$

Divide 20 by -1.

$$t^{2}-20t=-\frac{175}{-1}$$

Divide -175 by -1.

$$t^{2}-20t=175$$

Divide -20, the coefficient of the x term, by 2 to get -10. Then add the square of -10 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-20t+\left(-10\right)^{2}=175+\left(-10\right)^{2}$$

Square -10.

$$t^{2}-20t+100=175+100$$

Add 175 to 100.

$$t^{2}-20t+100=275$$

Factor t^{2}-20t+100. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-10\right)^{2}=275$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-10\right)^{2}}=\sqrt{275}$$

Simplify.

$$t-10=5\sqrt{11} t-10=-5\sqrt{11}$$

Add 10 to both sides of the equation.

$$t=5\sqrt{11}+10 t=10-5\sqrt{11}$$