Solve Q^2-60Q=-100

$$Q ^ { 2 } - 60 Q = - 100$$

Solve for Q

$$Q=20\sqrt{2}+30\approx 58.284271247$$

$$Q=30-20\sqrt{2}\approx 1.715728753$$

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.

$$Q^{2}-60Q=-100$$

Add 100 to both sides of the equation.

$$Q^{2}-60Q-\left(-100\right)=-100-\left(-100\right)$$

Subtracting -100 from itself leaves 0.

$$Q^{2}-60Q-\left(-100\right)=0$$

Subtract -100 from 0.

$$Q^{2}-60Q+100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -60 for b, and 100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$Q=\frac{-\left(-60\right)±\sqrt{\left(-60\right)^{2}-4\times 100}}{2}$$

Square -60.

$$Q=\frac{-\left(-60\right)±\sqrt{3600-4\times 100}}{2}$$

Multiply -4 times 100.

$$Q=\frac{-\left(-60\right)±\sqrt{3600-400}}{2}$$

Add 3600 to -400.

$$Q=\frac{-\left(-60\right)±\sqrt{3200}}{2}$$

Take the square root of 3200.

$$Q=\frac{-\left(-60\right)±40\sqrt{2}}{2}$$

The opposite of -60 is 60.

$$Q=\frac{60±40\sqrt{2}}{2}$$

Now solve the equation Q=\frac{60±40\sqrt{2}}{2} when ± is plus. Add 60 to 40\sqrt{2}.

$$Q=\frac{40\sqrt{2}+60}{2}$$

Divide 60+40\sqrt{2} by 2.

$$Q=20\sqrt{2}+30$$

Now solve the equation Q=\frac{60±40\sqrt{2}}{2} when ± is minus. Subtract 40\sqrt{2} from 60.

$$Q=\frac{60-40\sqrt{2}}{2}$$

Divide 60-40\sqrt{2} by 2.

$$Q=30-20\sqrt{2}$$

The equation is now solved.

$$Q=20\sqrt{2}+30 Q=30-20\sqrt{2}$$

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.

$$Q^{2}-60Q=-100$$

Divide -60, the coefficient of the x term, by 2 to get -30. Then add the square of -30 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$Q^{2}-60Q+\left(-30\right)^{2}=-100+\left(-30\right)^{2}$$

Square -30.

$$Q^{2}-60Q+900=-100+900$$

Add -100 to 900.

$$Q^{2}-60Q+900=800$$

Factor Q^{2}-60Q+900. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(Q-30\right)^{2}=800$$

Take the square root of both sides of the equation.

$$\sqrt{\left(Q-30\right)^{2}}=\sqrt{800}$$

Simplify.

$$Q-30=20\sqrt{2} Q-30=-20\sqrt{2}$$

Add 30 to both sides of the equation.

$$Q=20\sqrt{2}+30 Q=30-20\sqrt{2}$$