Solve 100=0.0065x^2-0.75x

$$100=0.0065 { x }^{ 2 } -0.75x$$

Solve for x

$$x = \frac{50 \sqrt{1265} + 750}{13} \approx 194.487840337$$

$$x=\frac{750-50\sqrt{1265}}{13}\approx -79.103224952$$

Steps Using the Quadratic Formula

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

$$0.0065x^{2}-0.75x=100$$

Subtract 100 from both sides.

$$0.0065x^{2}-0.75x-100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.0065 for a, -0.75 for b, and -100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-0.75\right)±\sqrt{\left(-0.75\right)^{2}-4\times 0.0065\left(-100\right)}}{2\times 0.0065}$$

Square -0.75 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-0.75\right)±\sqrt{0.5625-4\times 0.0065\left(-100\right)}}{2\times 0.0065}$$

Multiply -4 times 0.0065.

$$x=\frac{-\left(-0.75\right)±\sqrt{0.5625-0.026\left(-100\right)}}{2\times 0.0065}$$

Multiply -0.026 times -100.

$$x=\frac{-\left(-0.75\right)±\sqrt{0.5625+2.6}}{2\times 0.0065}$$

Add 0.5625 to 2.6 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-0.75\right)±\sqrt{3.1625}}{2\times 0.0065}$$

Take the square root of 3.1625.

$$x=\frac{-\left(-0.75\right)±\frac{\sqrt{1265}}{20}}{2\times 0.0065}$$

The opposite of -0.75 is 0.75.

$$x=\frac{0.75±\frac{\sqrt{1265}}{20}}{2\times 0.0065}$$

Multiply 2 times 0.0065.

$$x=\frac{0.75±\frac{\sqrt{1265}}{20}}{0.013}$$

Now solve the equation x=\frac{0.75±\frac{\sqrt{1265}}{20}}{0.013} when ± is plus. Add 0.75 to \frac{\sqrt{1265}}{20}.

$$x=\frac{\frac{\sqrt{1265}}{20}+\frac{3}{4}}{0.013}$$

Divide \frac{3}{4}+\frac{\sqrt{1265}}{20} by 0.013 by multiplying \frac{3}{4}+\frac{\sqrt{1265}}{20} by the reciprocal of 0.013.

$$x=\frac{50\sqrt{1265}+750}{13}$$

Now solve the equation x=\frac{0.75±\frac{\sqrt{1265}}{20}}{0.013} when ± is minus. Subtract \frac{\sqrt{1265}}{20} from 0.75.

$$x=\frac{-\frac{\sqrt{1265}}{20}+\frac{3}{4}}{0.013}$$

Divide \frac{3}{4}-\frac{\sqrt{1265}}{20} by 0.013 by multiplying \frac{3}{4}-\frac{\sqrt{1265}}{20} by the reciprocal of 0.013.

$$x=\frac{750-50\sqrt{1265}}{13}$$

The equation is now solved.

$$x=\frac{50\sqrt{1265}+750}{13} x=\frac{750-50\sqrt{1265}}{13}$$

Steps for Completing the Square

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

$$0.0065x^{2}-0.75x=100$$

Divide both sides of the equation by 0.0065, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{0.0065x^{2}-0.75x}{0.0065}=\frac{100}{0.0065}$$

Dividing by 0.0065 undoes the multiplication by 0.0065.

$$x^{2}+\left(-\frac{0.75}{0.0065}\right)x=\frac{100}{0.0065}$$

Divide -0.75 by 0.0065 by multiplying -0.75 by the reciprocal of 0.0065.

$$x^{2}-\frac{1500}{13}x=\frac{100}{0.0065}$$

Divide 100 by 0.0065 by multiplying 100 by the reciprocal of 0.0065.

$$x^{2}-\frac{1500}{13}x=\frac{200000}{13}$$

Divide -\frac{1500}{13}, the coefficient of the x term, by 2 to get -\frac{750}{13}. Then add the square of -\frac{750}{13} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1500}{13}x+\left(-\frac{750}{13}\right)^{2}=\frac{200000}{13}+\left(-\frac{750}{13}\right)^{2}$$

Square -\frac{750}{13} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1500}{13}x+\frac{562500}{169}=\frac{200000}{13}+\frac{562500}{169}$$

Add \frac{200000}{13} to \frac{562500}{169} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1500}{13}x+\frac{562500}{169}=\frac{3162500}{169}$$

Factor x^{2}-\frac{1500}{13}x+\frac{562500}{169}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{750}{13}\right)^{2}=\frac{3162500}{169}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{750}{13}\right)^{2}}=\sqrt{\frac{3162500}{169}}$$

Simplify.

$$x-\frac{750}{13}=\frac{50\sqrt{1265}}{13} x-\frac{750}{13}=-\frac{50\sqrt{1265}}{13}$$

Add \frac{750}{13} to both sides of the equation.

$$x=\frac{50\sqrt{1265}+750}{13} x=\frac{750-50\sqrt{1265}}{13}$$