Solve 0.05Q^2-30.1Q=5

$$0.05 Q ^ { 2 } - 30.1 Q = 5$$

Solve for Q

$$Q=\sqrt{90701}+301\approx 602.166067146$$

$$Q=301-\sqrt{90701}\approx -0.166067146$$

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.

$$0.05Q^{2}-30.1Q=5$$

Subtract 5 from both sides of the equation.

$$0.05Q^{2}-30.1Q-5=5-5$$

Subtracting 5 from itself leaves 0.

$$0.05Q^{2}-30.1Q-5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.05 for a, -30.1 for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$Q=\frac{-\left(-30.1\right)±\sqrt{\left(-30.1\right)^{2}-4\times 0.05\left(-5\right)}}{2\times 0.05}$$

Square -30.1 by squaring both the numerator and the denominator of the fraction.

$$Q=\frac{-\left(-30.1\right)±\sqrt{906.01-4\times 0.05\left(-5\right)}}{2\times 0.05}$$

Multiply -4 times 0.05.

$$Q=\frac{-\left(-30.1\right)±\sqrt{906.01-0.2\left(-5\right)}}{2\times 0.05}$$

Multiply -0.2 times -5.

$$Q=\frac{-\left(-30.1\right)±\sqrt{906.01+1}}{2\times 0.05}$$

Add 906.01 to 1.

$$Q=\frac{-\left(-30.1\right)±\sqrt{907.01}}{2\times 0.05}$$

Take the square root of 907.01.

$$Q=\frac{-\left(-30.1\right)±\frac{\sqrt{90701}}{10}}{2\times 0.05}$$

The opposite of -30.1 is 30.1.

$$Q=\frac{30.1±\frac{\sqrt{90701}}{10}}{2\times 0.05}$$

Multiply 2 times 0.05.

$$Q=\frac{30.1±\frac{\sqrt{90701}}{10}}{0.1}$$

Now solve the equation Q=\frac{30.1±\frac{\sqrt{90701}}{10}}{0.1} when ± is plus. Add 30.1 to \frac{\sqrt{90701}}{10}.

$$Q=\frac{\sqrt{90701}+301}{0.1\times 10}$$

Divide \frac{301+\sqrt{90701}}{10} by 0.1 by multiplying \frac{301+\sqrt{90701}}{10} by the reciprocal of 0.1.

$$Q=\sqrt{90701}+301$$

Now solve the equation Q=\frac{30.1±\frac{\sqrt{90701}}{10}}{0.1} when ± is minus. Subtract \frac{\sqrt{90701}}{10} from 30.1.

$$Q=\frac{301-\sqrt{90701}}{0.1\times 10}$$

Divide \frac{301-\sqrt{90701}}{10} by 0.1 by multiplying \frac{301-\sqrt{90701}}{10} by the reciprocal of 0.1.

$$Q=301-\sqrt{90701}$$

The equation is now solved.

$$Q=\sqrt{90701}+301 Q=301-\sqrt{90701}$$

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.

$$0.05Q^{2}-30.1Q=5$$

Multiply both sides by 20.

$$\frac{0.05Q^{2}-30.1Q}{0.05}=\frac{5}{0.05}$$

Dividing by 0.05 undoes the multiplication by 0.05.

$$Q^{2}+\left(-\frac{30.1}{0.05}\right)Q=\frac{5}{0.05}$$

Divide -30.1 by 0.05 by multiplying -30.1 by the reciprocal of 0.05.

$$Q^{2}-602Q=\frac{5}{0.05}$$

Divide 5 by 0.05 by multiplying 5 by the reciprocal of 0.05.

$$Q^{2}-602Q=100$$

Divide -602, the coefficient of the x term, by 2 to get -301. Then add the square of -301 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$Q^{2}-602Q+\left(-301\right)^{2}=100+\left(-301\right)^{2}$$

Square -301.

$$Q^{2}-602Q+90601=100+90601$$

Add 100 to 90601.

$$Q^{2}-602Q+90601=90701$$

Factor Q^{2}-602Q+90601. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(Q-301\right)^{2}=90701$$

Take the square root of both sides of the equation.

$$\sqrt{\left(Q-301\right)^{2}}=\sqrt{90701}$$

Simplify.

$$Q-301=\sqrt{90701} Q-301=-\sqrt{90701}$$

Add 301 to both sides of the equation.

$$Q=\sqrt{90701}+301 Q=301-\sqrt{90701}$$