Solve 81x^2-576x+237=0

$$81 { x }^{ 2 } -576x+237=0$$

Solve for x

$$x = \frac{\sqrt{787} + 32}{9} \approx 6.672613364$$

$$x=\frac{32-\sqrt{787}}{9}\approx 0.438497747$$

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.

$$81x^{2}-576x+237=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 81 for a, -576 for b, and 237 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-576\right)±\sqrt{\left(-576\right)^{2}-4\times 81\times 237}}{2\times 81}$$

Square -576.

$$x=\frac{-\left(-576\right)±\sqrt{331776-4\times 81\times 237}}{2\times 81}$$

Multiply -4 times 81.

$$x=\frac{-\left(-576\right)±\sqrt{331776-324\times 237}}{2\times 81}$$

Multiply -324 times 237.

$$x=\frac{-\left(-576\right)±\sqrt{331776-76788}}{2\times 81}$$

Add 331776 to -76788.

$$x=\frac{-\left(-576\right)±\sqrt{254988}}{2\times 81}$$

Take the square root of 254988.

$$x=\frac{-\left(-576\right)±18\sqrt{787}}{2\times 81}$$

The opposite of -576 is 576.

$$x=\frac{576±18\sqrt{787}}{2\times 81}$$

Multiply 2 times 81.

$$x=\frac{576±18\sqrt{787}}{162}$$

Now solve the equation x=\frac{576±18\sqrt{787}}{162} when ± is plus. Add 576 to 18\sqrt{787}.

$$x=\frac{18\sqrt{787}+576}{162}$$

Divide 576+18\sqrt{787} by 162.

$$x=\frac{\sqrt{787}+32}{9}$$

Now solve the equation x=\frac{576±18\sqrt{787}}{162} when ± is minus. Subtract 18\sqrt{787} from 576.

$$x=\frac{576-18\sqrt{787}}{162}$$

Divide 576-18\sqrt{787} by 162.

$$x=\frac{32-\sqrt{787}}{9}$$

The equation is now solved.

$$x=\frac{\sqrt{787}+32}{9} x=\frac{32-\sqrt{787}}{9}$$

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.

$$81x^{2}-576x+237=0$$

Subtract 237 from both sides of the equation.

$$81x^{2}-576x+237-237=-237$$

Subtracting 237 from itself leaves 0.

$$81x^{2}-576x=-237$$

Divide both sides by 81.

$$\frac{81x^{2}-576x}{81}=-\frac{237}{81}$$

Dividing by 81 undoes the multiplication by 81.

$$x^{2}+\left(-\frac{576}{81}\right)x=-\frac{237}{81}$$

Reduce the fraction \frac{-576}{81} to lowest terms by extracting and canceling out 9.

$$x^{2}-\frac{64}{9}x=-\frac{237}{81}$$

Reduce the fraction \frac{-237}{81} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{64}{9}x=-\frac{79}{27}$$

Divide -\frac{64}{9}, the coefficient of the x term, by 2 to get -\frac{32}{9}. Then add the square of -\frac{32}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{64}{9}x+\left(-\frac{32}{9}\right)^{2}=-\frac{79}{27}+\left(-\frac{32}{9}\right)^{2}$$

Square -\frac{32}{9} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{64}{9}x+\frac{1024}{81}=-\frac{79}{27}+\frac{1024}{81}$$

Add -\frac{79}{27} to \frac{1024}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{64}{9}x+\frac{1024}{81}=\frac{787}{81}$$

Factor x^{2}-\frac{64}{9}x+\frac{1024}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{32}{9}\right)^{2}=\frac{787}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{32}{9}\right)^{2}}=\sqrt{\frac{787}{81}}$$

Simplify.

$$x-\frac{32}{9}=\frac{\sqrt{787}}{9} x-\frac{32}{9}=-\frac{\sqrt{787}}{9}$$

Add \frac{32}{9} to both sides of the equation.

$$x=\frac{\sqrt{787}+32}{9} x=\frac{32-\sqrt{787}}{9}$$