Solve 70=-144.91x^2-20.865x+82.906

$$70=-144.91 { x }^{ 2 } -20.865x+82.906$$

Solve for x

$$x=\frac{3\sqrt{879575785}}{289820}-\frac{4173}{57964}\approx 0.23500079$$

$$x=-\frac{3\sqrt{879575785}}{289820}-\frac{4173}{57964}\approx -0.378986712$$

Steps Using the Quadratic Formula

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

$$-144.91x^{2}-20.865x+82.906=70$$

Subtract 70 from both sides.

$$-144.91x^{2}-20.865x+82.906-70=0$$

Subtract 70 from 82.906 to get 12.906.

$$-144.91x^{2}-20.865x+12.906=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -144.91 for a, -20.865 for b, and 12.906 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-20.865\right)±\sqrt{\left(-20.865\right)^{2}-4\left(-144.91\right)\times 12.906}}{2\left(-144.91\right)}$$

Square -20.865 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-20.865\right)±\sqrt{435.348225-4\left(-144.91\right)\times 12.906}}{2\left(-144.91\right)}$$

Multiply -4 times -144.91.

$$x=\frac{-\left(-20.865\right)±\sqrt{435.348225+579.64\times 12.906}}{2\left(-144.91\right)}$$

Multiply 579.64 times 12.906 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-20.865\right)±\sqrt{435.348225+7480.83384}}{2\left(-144.91\right)}$$

Add 435.348225 to 7480.83384 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-20.865\right)±\sqrt{7916.182065}}{2\left(-144.91\right)}$$

Take the square root of 7916.182065.

$$x=\frac{-\left(-20.865\right)±\frac{3\sqrt{879575785}}{1000}}{2\left(-144.91\right)}$$

The opposite of -20.865 is 20.865.

$$x=\frac{20.865±\frac{3\sqrt{879575785}}{1000}}{2\left(-144.91\right)}$$

Multiply 2 times -144.91.

$$x=\frac{20.865±\frac{3\sqrt{879575785}}{1000}}{-289.82}$$

Now solve the equation x=\frac{20.865±\frac{3\sqrt{879575785}}{1000}}{-289.82} when ± is plus. Add 20.865 to \frac{3\sqrt{879575785}}{1000}.

$$x=\frac{\frac{3\sqrt{879575785}}{1000}+\frac{4173}{200}}{-289.82}$$

Divide \frac{4173}{200}+\frac{3\sqrt{879575785}}{1000} by -289.82 by multiplying \frac{4173}{200}+\frac{3\sqrt{879575785}}{1000} by the reciprocal of -289.82.

$$x=-\frac{3\sqrt{879575785}}{289820}-\frac{4173}{57964}$$

Now solve the equation x=\frac{20.865±\frac{3\sqrt{879575785}}{1000}}{-289.82} when ± is minus. Subtract \frac{3\sqrt{879575785}}{1000} from 20.865.

$$x=\frac{-\frac{3\sqrt{879575785}}{1000}+\frac{4173}{200}}{-289.82}$$

Divide \frac{4173}{200}-\frac{3\sqrt{879575785}}{1000} by -289.82 by multiplying \frac{4173}{200}-\frac{3\sqrt{879575785}}{1000} by the reciprocal of -289.82.

$$x=\frac{3\sqrt{879575785}}{289820}-\frac{4173}{57964}$$

The equation is now solved.

$$x=-\frac{3\sqrt{879575785}}{289820}-\frac{4173}{57964} x=\frac{3\sqrt{879575785}}{289820}-\frac{4173}{57964}$$

Steps for Completing the Square

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

$$-144.91x^{2}-20.865x+82.906=70$$

Subtract 82.906 from both sides.

$$-144.91x^{2}-20.865x=70-82.906$$

Subtract 82.906 from 70 to get -12.906.

$$-144.91x^{2}-20.865x=-12.906$$

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.

$$-144.91x^{2}-20.865x=-\frac{6453}{500}$$

Divide both sides of the equation by -144.91, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-144.91x^{2}-20.865x}{-144.91}=-\frac{\frac{6453}{500}}{-144.91}$$

Dividing by -144.91 undoes the multiplication by -144.91.

$$x^{2}+\left(-\frac{20.865}{-144.91}\right)x=-\frac{\frac{6453}{500}}{-144.91}$$

Divide -20.865 by -144.91 by multiplying -20.865 by the reciprocal of -144.91.

$$x^{2}+\frac{4173}{28982}x=-\frac{\frac{6453}{500}}{-144.91}$$

Divide -\frac{6453}{500} by -144.91 by multiplying -\frac{6453}{500} by the reciprocal of -144.91.

$$x^{2}+\frac{4173}{28982}x=\frac{6453}{72455}$$

Divide \frac{4173}{28982}, the coefficient of the x term, by 2 to get \frac{4173}{57964}. Then add the square of \frac{4173}{57964} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{4173}{28982}x+\frac{4173}{57964}^{2}=\frac{6453}{72455}+\frac{4173}{57964}^{2}$$

Square \frac{4173}{57964} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{4173}{28982}x+\frac{17413929}{3359825296}=\frac{6453}{72455}+\frac{17413929}{3359825296}$$

Add \frac{6453}{72455} to \frac{17413929}{3359825296} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{4173}{28982}x+\frac{17413929}{3359825296}=\frac{1583236413}{16799126480}$$

Factor x^{2}+\frac{4173}{28982}x+\frac{17413929}{3359825296}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{4173}{57964}\right)^{2}=\frac{1583236413}{16799126480}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{4173}{57964}\right)^{2}}=\sqrt{\frac{1583236413}{16799126480}}$$

Simplify.

$$x+\frac{4173}{57964}=\frac{3\sqrt{879575785}}{289820} x+\frac{4173}{57964}=-\frac{3\sqrt{879575785}}{289820}$$

Subtract \frac{4173}{57964} from both sides of the equation.

$$x=\frac{3\sqrt{879575785}}{289820}-\frac{4173}{57964} x=-\frac{3\sqrt{879575785}}{289820}-\frac{4173}{57964}$$