Solve 116=7.65x^2-188.5x+1225.25

$$116 = 7.65 { x }^{ 2 } -188.5x+1225.25$$

Solve for x

$$x = \frac{2 \sqrt{39730} + 1885}{153} \approx 14.925802053$$

$$x = \frac{1885 - 2 \sqrt{39730}}{153} \approx 9.714720823$$

Steps Using the Quadratic Formula

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

$$7.65x^{2}-188.5x+1225.25=116$$

Subtract 116 from both sides.

$$7.65x^{2}-188.5x+1225.25-116=0$$

Subtract 116 from 1225.25 to get 1109.25.

$$7.65x^{2}-188.5x+1109.25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7.65 for a, -188.5 for b, and 1109.25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-188.5\right)±\sqrt{\left(-188.5\right)^{2}-4\times 7.65\times 1109.25}}{2\times 7.65}$$

Square -188.5 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-188.5\right)±\sqrt{35532.25-4\times 7.65\times 1109.25}}{2\times 7.65}$$

Multiply -4 times 7.65.

$$x=\frac{-\left(-188.5\right)±\sqrt{35532.25-30.6\times 1109.25}}{2\times 7.65}$$

Multiply -30.6 times 1109.25 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-188.5\right)±\sqrt{35532.25-33943.05}}{2\times 7.65}$$

Add 35532.25 to -33943.05 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-188.5\right)±\sqrt{1589.2}}{2\times 7.65}$$

Take the square root of 1589.2.

$$x=\frac{-\left(-188.5\right)±\frac{\sqrt{39730}}{5}}{2\times 7.65}$$

The opposite of -188.5 is 188.5.

$$x=\frac{188.5±\frac{\sqrt{39730}}{5}}{2\times 7.65}$$

Multiply 2 times 7.65.

$$x=\frac{188.5±\frac{\sqrt{39730}}{5}}{15.3}$$

Now solve the equation x=\frac{188.5±\frac{\sqrt{39730}}{5}}{15.3} when ± is plus. Add 188.5 to \frac{\sqrt{39730}}{5}.

$$x=\frac{\frac{\sqrt{39730}}{5}+\frac{377}{2}}{15.3}$$

Divide \frac{377}{2}+\frac{\sqrt{39730}}{5} by 15.3 by multiplying \frac{377}{2}+\frac{\sqrt{39730}}{5} by the reciprocal of 15.3.

$$x=\frac{2\sqrt{39730}+1885}{153}$$

Now solve the equation x=\frac{188.5±\frac{\sqrt{39730}}{5}}{15.3} when ± is minus. Subtract \frac{\sqrt{39730}}{5} from 188.5.

$$x=\frac{-\frac{\sqrt{39730}}{5}+\frac{377}{2}}{15.3}$$

Divide \frac{377}{2}-\frac{\sqrt{39730}}{5} by 15.3 by multiplying \frac{377}{2}-\frac{\sqrt{39730}}{5} by the reciprocal of 15.3.

$$x=\frac{1885-2\sqrt{39730}}{153}$$

The equation is now solved.

$$x=\frac{2\sqrt{39730}+1885}{153} x=\frac{1885-2\sqrt{39730}}{153}$$

Steps for Completing the Square

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

$$7.65x^{2}-188.5x+1225.25=116$$

Subtract 1225.25 from both sides.

$$7.65x^{2}-188.5x=116-1225.25$$

Subtract 1225.25 from 116 to get -1109.25.

$$7.65x^{2}-188.5x=-1109.25$$

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.

$$7.65x^{2}-188.5x=-\frac{4437}{4}$$

Divide both sides of the equation by 7.65, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{7.65x^{2}-188.5x}{7.65}=-\frac{\frac{4437}{4}}{7.65}$$

Dividing by 7.65 undoes the multiplication by 7.65.

$$x^{2}+\left(-\frac{188.5}{7.65}\right)x=-\frac{\frac{4437}{4}}{7.65}$$

Divide -188.5 by 7.65 by multiplying -188.5 by the reciprocal of 7.65.

$$x^{2}-\frac{3770}{153}x=-\frac{\frac{4437}{4}}{7.65}$$

Divide -\frac{4437}{4} by 7.65 by multiplying -\frac{4437}{4} by the reciprocal of 7.65.

$$x^{2}-\frac{3770}{153}x=-145$$

Divide -\frac{3770}{153}, the coefficient of the x term, by 2 to get -\frac{1885}{153}. Then add the square of -\frac{1885}{153} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{3770}{153}x+\left(-\frac{1885}{153}\right)^{2}=-145+\left(-\frac{1885}{153}\right)^{2}$$

Square -\frac{1885}{153} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3770}{153}x+\frac{3553225}{23409}=-145+\frac{3553225}{23409}$$

Add -145 to \frac{3553225}{23409}.

$$x^{2}-\frac{3770}{153}x+\frac{3553225}{23409}=\frac{158920}{23409}$$

Factor x^{2}-\frac{3770}{153}x+\frac{3553225}{23409}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1885}{153}\right)^{2}=\frac{158920}{23409}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1885}{153}\right)^{2}}=\sqrt{\frac{158920}{23409}}$$

Simplify.

$$x-\frac{1885}{153}=\frac{2\sqrt{39730}}{153} x-\frac{1885}{153}=-\frac{2\sqrt{39730}}{153}$$

Add \frac{1885}{153} to both sides of the equation.

$$x=\frac{2\sqrt{39730}+1885}{153} x=\frac{1885-2\sqrt{39730}}{153}$$