Solve (-2x+9)(-9x+5)+(-9x-5)^2=0

$$( - 2 x + 9 ) ( - 9 x + 5 ) + ( - 9 x - 5 ) ^ { 2 } = 0$$

Solve for x (complex solution)

$$x=\frac{1+\sqrt{27719}i}{198}\approx 0.005050505+0.840859798i$$

$$x=\frac{-\sqrt{27719}i+1}{198}\approx 0.005050505-0.840859798i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply -2x+9 by -9x+5 and combine like terms.

$$18x^{2}-91x+45+\left(-9x-5\right)^{2}=0$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-9x-5\right)^{2}.

$$18x^{2}-91x+45+81x^{2}+90x+25=0$$

Combine 18x^{2} and 81x^{2} to get 99x^{2}.

$$99x^{2}-91x+45+90x+25=0$$

Combine -91x and 90x to get -x.

$$99x^{2}-x+45+25=0$$

Add 45 and 25 to get 70.

$$99x^{2}-x+70=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 99 for a, -1 for b, and 70 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 99\times 70}}{2\times 99}$$

Multiply -4 times 99.

$$x=\frac{-\left(-1\right)±\sqrt{1-396\times 70}}{2\times 99}$$

Multiply -396 times 70.

$$x=\frac{-\left(-1\right)±\sqrt{1-27720}}{2\times 99}$$

Add 1 to -27720.

$$x=\frac{-\left(-1\right)±\sqrt{-27719}}{2\times 99}$$

Take the square root of -27719.

$$x=\frac{-\left(-1\right)±\sqrt{27719}i}{2\times 99}$$

The opposite of -1 is 1.

$$x=\frac{1±\sqrt{27719}i}{2\times 99}$$

Multiply 2 times 99.

$$x=\frac{1±\sqrt{27719}i}{198}$$

Now solve the equation x=\frac{1±\sqrt{27719}i}{198} when ± is plus. Add 1 to i\sqrt{27719}.

$$x=\frac{1+\sqrt{27719}i}{198}$$

Now solve the equation x=\frac{1±\sqrt{27719}i}{198} when ± is minus. Subtract i\sqrt{27719} from 1.

$$x=\frac{-\sqrt{27719}i+1}{198}$$

The equation is now solved.

$$x=\frac{1+\sqrt{27719}i}{198} x=\frac{-\sqrt{27719}i+1}{198}$$

Steps for Completing the Square

Use the distributive property to multiply -2x+9 by -9x+5 and combine like terms.

$$18x^{2}-91x+45+\left(-9x-5\right)^{2}=0$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-9x-5\right)^{2}.

$$18x^{2}-91x+45+81x^{2}+90x+25=0$$

Combine 18x^{2} and 81x^{2} to get 99x^{2}.

$$99x^{2}-91x+45+90x+25=0$$

Combine -91x and 90x to get -x.

$$99x^{2}-x+45+25=0$$

Add 45 and 25 to get 70.

$$99x^{2}-x+70=0$$

Subtract 70 from both sides. Anything subtracted from zero gives its negation.

$$99x^{2}-x=-70$$

Divide both sides by 99.

$$\frac{99x^{2}-x}{99}=-\frac{70}{99}$$

Dividing by 99 undoes the multiplication by 99.

$$x^{2}-\frac{1}{99}x=-\frac{70}{99}$$

Divide -\frac{1}{99}, the coefficient of the x term, by 2 to get -\frac{1}{198}. Then add the square of -\frac{1}{198} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{99}x+\left(-\frac{1}{198}\right)^{2}=-\frac{70}{99}+\left(-\frac{1}{198}\right)^{2}$$

Square -\frac{1}{198} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{99}x+\frac{1}{39204}=-\frac{70}{99}+\frac{1}{39204}$$

Add -\frac{70}{99} to \frac{1}{39204} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1}{99}x+\frac{1}{39204}=-\frac{27719}{39204}$$

Factor x^{2}-\frac{1}{99}x+\frac{1}{39204}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{198}\right)^{2}=-\frac{27719}{39204}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{198}\right)^{2}}=\sqrt{-\frac{27719}{39204}}$$

Simplify.

$$x-\frac{1}{198}=\frac{\sqrt{27719}i}{198} x-\frac{1}{198}=-\frac{\sqrt{27719}i}{198}$$

Add \frac{1}{198} to both sides of the equation.

$$x=\frac{1+\sqrt{27719}i}{198} x=\frac{-\sqrt{27719}i+1}{198}$$