Solve (2x+9)(2x-9)=9x^2-8x

$$( 2 x + 9 ) ( 2 x - 9 ) = 9 x ^ { 2 } - 8 x$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{389}i+4}{5}\approx 0.8-3.944616585i$$

$$x=\frac{4+\sqrt{389}i}{5}\approx 0.8+3.944616585i$$

Steps Using the Quadratic Formula

Consider \left(2x+9\right)\left(2x-9\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 9.

$$\left(2x\right)^{2}-81=9x^{2}-8x$$

Expand \left(2x\right)^{2}.

$$2^{2}x^{2}-81=9x^{2}-8x$$

Calculate 2 to the power of 2 and get 4.

$$4x^{2}-81=9x^{2}-8x$$

Subtract 9x^{2} from both sides.

$$4x^{2}-81-9x^{2}=-8x$$

Combine 4x^{2} and -9x^{2} to get -5x^{2}.

$$-5x^{2}-81=-8x$$

Add 8x to both sides.

$$-5x^{2}-81+8x=0$$

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.

$$-5x^{2}+8x-81=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 8 for b, and -81 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-5\right)\left(-81\right)}}{2\left(-5\right)}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\left(-5\right)\left(-81\right)}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-8±\sqrt{64+20\left(-81\right)}}{2\left(-5\right)}$$

Multiply 20 times -81.

$$x=\frac{-8±\sqrt{64-1620}}{2\left(-5\right)}$$

Add 64 to -1620.

$$x=\frac{-8±\sqrt{-1556}}{2\left(-5\right)}$$

Take the square root of -1556.

$$x=\frac{-8±2\sqrt{389}i}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{-8±2\sqrt{389}i}{-10}$$

Now solve the equation x=\frac{-8±2\sqrt{389}i}{-10} when ± is plus. Add -8 to 2i\sqrt{389}.

$$x=\frac{-8+2\sqrt{389}i}{-10}$$

Divide -8+2i\sqrt{389} by -10.

$$x=\frac{-\sqrt{389}i+4}{5}$$

Now solve the equation x=\frac{-8±2\sqrt{389}i}{-10} when ± is minus. Subtract 2i\sqrt{389} from -8.

$$x=\frac{-2\sqrt{389}i-8}{-10}$$

Divide -8-2i\sqrt{389} by -10.

$$x=\frac{4+\sqrt{389}i}{5}$$

The equation is now solved.

$$x=\frac{-\sqrt{389}i+4}{5} x=\frac{4+\sqrt{389}i}{5}$$

Steps for Completing the Square

Consider \left(2x+9\right)\left(2x-9\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 9.

$$\left(2x\right)^{2}-81=9x^{2}-8x$$

Expand \left(2x\right)^{2}.

$$2^{2}x^{2}-81=9x^{2}-8x$$

Calculate 2 to the power of 2 and get 4.

$$4x^{2}-81=9x^{2}-8x$$

Subtract 9x^{2} from both sides.

$$4x^{2}-81-9x^{2}=-8x$$

Combine 4x^{2} and -9x^{2} to get -5x^{2}.

$$-5x^{2}-81=-8x$$

Add 8x to both sides.

$$-5x^{2}-81+8x=0$$

Add 81 to both sides. Anything plus zero gives itself.

$$-5x^{2}+8x=81$$

Divide both sides by -5.

$$\frac{-5x^{2}+8x}{-5}=\frac{81}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$x^{2}+\frac{8}{-5}x=\frac{81}{-5}$$

Divide 8 by -5.

$$x^{2}-\frac{8}{5}x=\frac{81}{-5}$$

Divide 81 by -5.

$$x^{2}-\frac{8}{5}x=-\frac{81}{5}$$

Divide -\frac{8}{5}, the coefficient of the x term, by 2 to get -\frac{4}{5}. Then add the square of -\frac{4}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{8}{5}x+\left(-\frac{4}{5}\right)^{2}=-\frac{81}{5}+\left(-\frac{4}{5}\right)^{2}$$

Square -\frac{4}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{8}{5}x+\frac{16}{25}=-\frac{81}{5}+\frac{16}{25}$$

Add -\frac{81}{5} to \frac{16}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{8}{5}x+\frac{16}{25}=-\frac{389}{25}$$

Factor x^{2}-\frac{8}{5}x+\frac{16}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{4}{5}\right)^{2}=-\frac{389}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{4}{5}\right)^{2}}=\sqrt{-\frac{389}{25}}$$

Simplify.

$$x-\frac{4}{5}=\frac{\sqrt{389}i}{5} x-\frac{4}{5}=-\frac{\sqrt{389}i}{5}$$

Add \frac{4}{5} to both sides of the equation.

$$x=\frac{4+\sqrt{389}i}{5} x=\frac{-\sqrt{389}i+4}{5}$$