Solve 36x^2-81x+36=0

$$36 { x }^{ 2 } -81x+36=0$$

Solve for x

$$x = \frac{\sqrt{17} + 9}{8} \approx 1.640388203$$

$$x=\frac{9-\sqrt{17}}{8}\approx 0.609611797$$

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.

$$36x^{2}-81x+36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 36 for a, -81 for b, and 36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-81\right)±\sqrt{\left(-81\right)^{2}-4\times 36\times 36}}{2\times 36}$$

Square -81.

$$x=\frac{-\left(-81\right)±\sqrt{6561-4\times 36\times 36}}{2\times 36}$$

Multiply -4 times 36.

$$x=\frac{-\left(-81\right)±\sqrt{6561-144\times 36}}{2\times 36}$$

Multiply -144 times 36.

$$x=\frac{-\left(-81\right)±\sqrt{6561-5184}}{2\times 36}$$

Add 6561 to -5184.

$$x=\frac{-\left(-81\right)±\sqrt{1377}}{2\times 36}$$

Take the square root of 1377.

$$x=\frac{-\left(-81\right)±9\sqrt{17}}{2\times 36}$$

The opposite of -81 is 81.

$$x=\frac{81±9\sqrt{17}}{2\times 36}$$

Multiply 2 times 36.

$$x=\frac{81±9\sqrt{17}}{72}$$

Now solve the equation x=\frac{81±9\sqrt{17}}{72} when ± is plus. Add 81 to 9\sqrt{17}.

$$x=\frac{9\sqrt{17}+81}{72}$$

Divide 81+9\sqrt{17} by 72.

$$x=\frac{\sqrt{17}+9}{8}$$

Now solve the equation x=\frac{81±9\sqrt{17}}{72} when ± is minus. Subtract 9\sqrt{17} from 81.

$$x=\frac{81-9\sqrt{17}}{72}$$

Divide 81-9\sqrt{17} by 72.

$$x=\frac{9-\sqrt{17}}{8}$$

The equation is now solved.

$$x=\frac{\sqrt{17}+9}{8} x=\frac{9-\sqrt{17}}{8}$$

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.

$$36x^{2}-81x+36=0$$

Subtract 36 from both sides of the equation.

$$36x^{2}-81x+36-36=-36$$

Subtracting 36 from itself leaves 0.

$$36x^{2}-81x=-36$$

Divide both sides by 36.

$$\frac{36x^{2}-81x}{36}=-\frac{36}{36}$$

Dividing by 36 undoes the multiplication by 36.

$$x^{2}+\left(-\frac{81}{36}\right)x=-\frac{36}{36}$$

Reduce the fraction \frac{-81}{36} to lowest terms by extracting and canceling out 9.

$$x^{2}-\frac{9}{4}x=-\frac{36}{36}$$

Divide -36 by 36.

$$x^{2}-\frac{9}{4}x=-1$$

Divide -\frac{9}{4}, the coefficient of the x term, by 2 to get -\frac{9}{8}. Then add the square of -\frac{9}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{9}{4}x+\left(-\frac{9}{8}\right)^{2}=-1+\left(-\frac{9}{8}\right)^{2}$$

Square -\frac{9}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{9}{4}x+\frac{81}{64}=-1+\frac{81}{64}$$

Add -1 to \frac{81}{64}.

$$x^{2}-\frac{9}{4}x+\frac{81}{64}=\frac{17}{64}$$

Factor x^{2}-\frac{9}{4}x+\frac{81}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9}{8}\right)^{2}=\frac{17}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{8}\right)^{2}}=\sqrt{\frac{17}{64}}$$

Simplify.

$$x-\frac{9}{8}=\frac{\sqrt{17}}{8} x-\frac{9}{8}=-\frac{\sqrt{17}}{8}$$

Add \frac{9}{8} to both sides of the equation.

$$x=\frac{\sqrt{17}+9}{8} x=\frac{9-\sqrt{17}}{8}$$