Solve 0.01q^2+0.30q-9775=0

$$0.01 q ^ { 2 } + 0.30 q - 9775 = 0$$

Solve for q

$$q=5\sqrt{39109}-15\approx 973.799777508$$

$$q=-5\sqrt{39109}-15\approx -1003.799777508$$

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.

$$0.01q^{2}+0.3q-9775=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.01 for a, 0.3 for b, and -9775 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$q=\frac{-0.3±\sqrt{0.3^{2}-4\times 0.01\left(-9775\right)}}{2\times 0.01}$$

Square 0.3 by squaring both the numerator and the denominator of the fraction.

$$q=\frac{-0.3±\sqrt{0.09-4\times 0.01\left(-9775\right)}}{2\times 0.01}$$

Multiply -4 times 0.01.

$$q=\frac{-0.3±\sqrt{0.09-0.04\left(-9775\right)}}{2\times 0.01}$$

Multiply -0.04 times -9775.

$$q=\frac{-0.3±\sqrt{0.09+391}}{2\times 0.01}$$

Add 0.09 to 391.

$$q=\frac{-0.3±\sqrt{391.09}}{2\times 0.01}$$

Take the square root of 391.09.

$$q=\frac{-0.3±\frac{\sqrt{39109}}{10}}{2\times 0.01}$$

Multiply 2 times 0.01.

$$q=\frac{-0.3±\frac{\sqrt{39109}}{10}}{0.02}$$

Now solve the equation q=\frac{-0.3±\frac{\sqrt{39109}}{10}}{0.02} when ± is plus. Add -0.3 to \frac{\sqrt{39109}}{10}.

$$q=\frac{\sqrt{39109}-3}{0.02\times 10}$$

Divide \frac{-3+\sqrt{39109}}{10} by 0.02 by multiplying \frac{-3+\sqrt{39109}}{10} by the reciprocal of 0.02.

$$q=5\sqrt{39109}-15$$

Now solve the equation q=\frac{-0.3±\frac{\sqrt{39109}}{10}}{0.02} when ± is minus. Subtract \frac{\sqrt{39109}}{10} from -0.3.

$$q=\frac{-\sqrt{39109}-3}{0.02\times 10}$$

Divide \frac{-3-\sqrt{39109}}{10} by 0.02 by multiplying \frac{-3-\sqrt{39109}}{10} by the reciprocal of 0.02.

$$q=-5\sqrt{39109}-15$$

The equation is now solved.

$$q=5\sqrt{39109}-15 q=-5\sqrt{39109}-15$$

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.

$$0.01q^{2}+0.3q-9775=0$$

Add 9775 to both sides of the equation.

$$0.01q^{2}+0.3q-9775-\left(-9775\right)=-\left(-9775\right)$$

Subtracting -9775 from itself leaves 0.

$$0.01q^{2}+0.3q=-\left(-9775\right)$$

Subtract -9775 from 0.

$$0.01q^{2}+0.3q=9775$$

Multiply both sides by 100.

$$\frac{0.01q^{2}+0.3q}{0.01}=\frac{9775}{0.01}$$

Dividing by 0.01 undoes the multiplication by 0.01.

$$q^{2}+\frac{0.3}{0.01}q=\frac{9775}{0.01}$$

Divide 0.3 by 0.01 by multiplying 0.3 by the reciprocal of 0.01.

$$q^{2}+30q=\frac{9775}{0.01}$$

Divide 9775 by 0.01 by multiplying 9775 by the reciprocal of 0.01.

$$q^{2}+30q=977500$$

Divide 30, the coefficient of the x term, by 2 to get 15. Then add the square of 15 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$q^{2}+30q+15^{2}=977500+15^{2}$$

Square 15.

$$q^{2}+30q+225=977500+225$$

Add 977500 to 225.

$$q^{2}+30q+225=977725$$

Factor q^{2}+30q+225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(q+15\right)^{2}=977725$$

Take the square root of both sides of the equation.

$$\sqrt{\left(q+15\right)^{2}}=\sqrt{977725}$$

Simplify.

$$q+15=5\sqrt{39109} q+15=-5\sqrt{39109}$$

Subtract 15 from both sides of the equation.

$$q=5\sqrt{39109}-15 q=-5\sqrt{39109}-15$$