Solve 1/125x^2-2/25x-24/5-0.48(x-29)=1.8

$$\frac{ 1 }{ 125 } { x }^{ 2 } - \frac{ 2 }{ 25 } x- \frac{ 24 }{ 5 } -0.48(x-29)=1.8$$

Solve for x

$$x=\sqrt{310}+35\approx 52.606816862$$

$$x=35-\sqrt{310}\approx 17.393183138$$

Steps Using the Quadratic Formula

Use the distributive property to multiply -0.48 by x-29.

$$\frac{1}{125}x^{2}-\frac{2}{25}x-\frac{24}{5}-0.48x+13.92=1.8$$

Combine -\frac{2}{25}x and -0.48x to get -\frac{14}{25}x.

$$\frac{1}{125}x^{2}-\frac{14}{25}x-\frac{24}{5}+13.92=1.8$$

Add -\frac{24}{5} and 13.92 to get \frac{228}{25}.

$$\frac{1}{125}x^{2}-\frac{14}{25}x+\frac{228}{25}=1.8$$

Subtract 1.8 from both sides.

$$\frac{1}{125}x^{2}-\frac{14}{25}x+\frac{228}{25}-1.8=0$$

Subtract 1.8 from \frac{228}{25} to get \frac{183}{25}.

$$\frac{1}{125}x^{2}-\frac{14}{25}x+\frac{183}{25}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{125} for a, -\frac{14}{25} for b, and \frac{183}{25} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{14}{25}\right)±\sqrt{\left(-\frac{14}{25}\right)^{2}-4\times \frac{1}{125}\times \frac{183}{25}}}{2\times \frac{1}{125}}$$

Square -\frac{14}{25} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{14}{25}\right)±\sqrt{\frac{196}{625}-4\times \frac{1}{125}\times \frac{183}{25}}}{2\times \frac{1}{125}}$$

Multiply -4 times \frac{1}{125}.

$$x=\frac{-\left(-\frac{14}{25}\right)±\sqrt{\frac{196}{625}-\frac{4}{125}\times \frac{183}{25}}}{2\times \frac{1}{125}}$$

Multiply -\frac{4}{125} times \frac{183}{25} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{14}{25}\right)±\sqrt{\frac{196}{625}-\frac{732}{3125}}}{2\times \frac{1}{125}}$$

Add \frac{196}{625} to -\frac{732}{3125} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{14}{25}\right)±\sqrt{\frac{248}{3125}}}{2\times \frac{1}{125}}$$

Take the square root of \frac{248}{3125}.

$$x=\frac{-\left(-\frac{14}{25}\right)±\frac{2\sqrt{310}}{125}}{2\times \frac{1}{125}}$$

The opposite of -\frac{14}{25} is \frac{14}{25}.

$$x=\frac{\frac{14}{25}±\frac{2\sqrt{310}}{125}}{2\times \frac{1}{125}}$$

Multiply 2 times \frac{1}{125}.

$$x=\frac{\frac{14}{25}±\frac{2\sqrt{310}}{125}}{\frac{2}{125}}$$

Now solve the equation x=\frac{\frac{14}{25}±\frac{2\sqrt{310}}{125}}{\frac{2}{125}} when ± is plus. Add \frac{14}{25} to \frac{2\sqrt{310}}{125}.

$$x=\frac{\frac{2\sqrt{310}}{125}+\frac{14}{25}}{\frac{2}{125}}$$

Divide \frac{14}{25}+\frac{2\sqrt{310}}{125} by \frac{2}{125} by multiplying \frac{14}{25}+\frac{2\sqrt{310}}{125} by the reciprocal of \frac{2}{125}.

$$x=\sqrt{310}+35$$

Now solve the equation x=\frac{\frac{14}{25}±\frac{2\sqrt{310}}{125}}{\frac{2}{125}} when ± is minus. Subtract \frac{2\sqrt{310}}{125} from \frac{14}{25}.

$$x=\frac{-\frac{2\sqrt{310}}{125}+\frac{14}{25}}{\frac{2}{125}}$$

Divide \frac{14}{25}-\frac{2\sqrt{310}}{125} by \frac{2}{125} by multiplying \frac{14}{25}-\frac{2\sqrt{310}}{125} by the reciprocal of \frac{2}{125}.

$$x=35-\sqrt{310}$$

The equation is now solved.

$$x=\sqrt{310}+35 x=35-\sqrt{310}$$

Steps for Completing the Square

Use the distributive property to multiply -0.48 by x-29.

$$\frac{1}{125}x^{2}-\frac{2}{25}x-\frac{24}{5}-0.48x+13.92=1.8$$

Combine -\frac{2}{25}x and -0.48x to get -\frac{14}{25}x.

$$\frac{1}{125}x^{2}-\frac{14}{25}x-\frac{24}{5}+13.92=1.8$$

Add -\frac{24}{5} and 13.92 to get \frac{228}{25}.

$$\frac{1}{125}x^{2}-\frac{14}{25}x+\frac{228}{25}=1.8$$

Subtract \frac{228}{25} from both sides.

$$\frac{1}{125}x^{2}-\frac{14}{25}x=1.8-\frac{228}{25}$$

Subtract \frac{228}{25} from 1.8 to get -\frac{183}{25}.

$$\frac{1}{125}x^{2}-\frac{14}{25}x=-\frac{183}{25}$$

Multiply both sides by 125.

$$\frac{\frac{1}{125}x^{2}-\frac{14}{25}x}{\frac{1}{125}}=-\frac{\frac{183}{25}}{\frac{1}{125}}$$

Dividing by \frac{1}{125} undoes the multiplication by \frac{1}{125}.

$$x^{2}+\left(-\frac{\frac{14}{25}}{\frac{1}{125}}\right)x=-\frac{\frac{183}{25}}{\frac{1}{125}}$$

Divide -\frac{14}{25} by \frac{1}{125} by multiplying -\frac{14}{25} by the reciprocal of \frac{1}{125}.

$$x^{2}-70x=-\frac{\frac{183}{25}}{\frac{1}{125}}$$

Divide -\frac{183}{25} by \frac{1}{125} by multiplying -\frac{183}{25} by the reciprocal of \frac{1}{125}.

$$x^{2}-70x=-915$$

Divide -70, the coefficient of the x term, by 2 to get -35. Then add the square of -35 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-70x+\left(-35\right)^{2}=-915+\left(-35\right)^{2}$$

Square -35.

$$x^{2}-70x+1225=-915+1225$$

Add -915 to 1225.

$$x^{2}-70x+1225=310$$

Factor x^{2}-70x+1225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-35\right)^{2}=310$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-35\right)^{2}}=\sqrt{310}$$

Simplify.

$$x-35=\sqrt{310} x-35=-\sqrt{310}$$

Add 35 to both sides of the equation.

$$x=\sqrt{310}+35 x=35-\sqrt{310}$$