Solve 5x^2+4x(15-5x)+9=120

$$5 { x }^{ 2 } +4x(15-5x)+9=120$$

Solve for x (complex solution)

$$x=\frac{\sqrt{85}i}{5}+2\approx 2+1.843908891i$$

$$x=-\frac{\sqrt{85}i}{5}+2\approx 2-1.843908891i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 4x by 15-5x.

$$5x^{2}+60x-20x^{2}+9=120$$

Combine 5x^{2} and -20x^{2} to get -15x^{2}.

$$-15x^{2}+60x+9=120$$

Subtract 120 from both sides.

$$-15x^{2}+60x+9-120=0$$

Subtract 120 from 9 to get -111.

$$-15x^{2}+60x-111=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -15 for a, 60 for b, and -111 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-60±\sqrt{60^{2}-4\left(-15\right)\left(-111\right)}}{2\left(-15\right)}$$

Square 60.

$$x=\frac{-60±\sqrt{3600-4\left(-15\right)\left(-111\right)}}{2\left(-15\right)}$$

Multiply -4 times -15.

$$x=\frac{-60±\sqrt{3600+60\left(-111\right)}}{2\left(-15\right)}$$

Multiply 60 times -111.

$$x=\frac{-60±\sqrt{3600-6660}}{2\left(-15\right)}$$

Add 3600 to -6660.

$$x=\frac{-60±\sqrt{-3060}}{2\left(-15\right)}$$

Take the square root of -3060.

$$x=\frac{-60±6\sqrt{85}i}{2\left(-15\right)}$$

Multiply 2 times -15.

$$x=\frac{-60±6\sqrt{85}i}{-30}$$

Now solve the equation x=\frac{-60±6\sqrt{85}i}{-30} when ± is plus. Add -60 to 6i\sqrt{85}.

$$x=\frac{-60+6\sqrt{85}i}{-30}$$

Divide -60+6i\sqrt{85} by -30.

$$x=-\frac{\sqrt{85}i}{5}+2$$

Now solve the equation x=\frac{-60±6\sqrt{85}i}{-30} when ± is minus. Subtract 6i\sqrt{85} from -60.

$$x=\frac{-6\sqrt{85}i-60}{-30}$$

Divide -60-6i\sqrt{85} by -30.

$$x=\frac{\sqrt{85}i}{5}+2$$

The equation is now solved.

$$x=-\frac{\sqrt{85}i}{5}+2 x=\frac{\sqrt{85}i}{5}+2$$

Steps for Completing the Square

Use the distributive property to multiply 4x by 15-5x.

$$5x^{2}+60x-20x^{2}+9=120$$

Combine 5x^{2} and -20x^{2} to get -15x^{2}.

$$-15x^{2}+60x+9=120$$

Subtract 9 from both sides.

$$-15x^{2}+60x=120-9$$

Subtract 9 from 120 to get 111.

$$-15x^{2}+60x=111$$

Divide both sides by -15.

$$\frac{-15x^{2}+60x}{-15}=\frac{111}{-15}$$

Dividing by -15 undoes the multiplication by -15.

$$x^{2}+\frac{60}{-15}x=\frac{111}{-15}$$

Divide 60 by -15.

$$x^{2}-4x=\frac{111}{-15}$$

Reduce the fraction \frac{111}{-15} to lowest terms by extracting and canceling out 3.

$$x^{2}-4x=-\frac{37}{5}$$

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

$$x^{2}-4x+\left(-2\right)^{2}=-\frac{37}{5}+\left(-2\right)^{2}$$

Square -2.

$$x^{2}-4x+4=-\frac{37}{5}+4$$

Add -\frac{37}{5} to 4.

$$x^{2}-4x+4=-\frac{17}{5}$$

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

$$\left(x-2\right)^{2}=-\frac{17}{5}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{-\frac{17}{5}}$$

Simplify.

$$x-2=\frac{\sqrt{85}i}{5} x-2=-\frac{\sqrt{85}i}{5}$$

Add 2 to both sides of the equation.

$$x=\frac{\sqrt{85}i}{5}+2 x=-\frac{\sqrt{85}i}{5}+2$$