Solve 6x=0.05*(2x+12)*(2x+18)

$$6x=0.05 \times (2x+12) \times (2x+18)$$

Solve for x

$$x=6$$

$$x=9$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 0.05 by 2x+12.

$$6x=\left(0.1x+0.6\right)\left(2x+18\right)$$

Use the distributive property to multiply 0.1x+0.6 by 2x+18 and combine like terms.

$$6x=0.2x^{2}+3x+10.8$$

Subtract 0.2x^{2} from both sides.

$$6x-0.2x^{2}=3x+10.8$$

Subtract 3x from both sides.

$$6x-0.2x^{2}-3x=10.8$$

Combine 6x and -3x to get 3x.

$$3x-0.2x^{2}=10.8$$

Subtract 10.8 from both sides.

$$3x-0.2x^{2}-10.8=0$$

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.

$$-0.2x^{2}+3x-10.8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -0.2 for a, 3 for b, and -10.8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\left(-0.2\right)\left(-10.8\right)}}{2\left(-0.2\right)}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\left(-0.2\right)\left(-10.8\right)}}{2\left(-0.2\right)}$$

Multiply -4 times -0.2.

$$x=\frac{-3±\sqrt{9+0.8\left(-10.8\right)}}{2\left(-0.2\right)}$$

Multiply 0.8 times -10.8 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-3±\sqrt{9-8.64}}{2\left(-0.2\right)}$$

Add 9 to -8.64.

$$x=\frac{-3±\sqrt{0.36}}{2\left(-0.2\right)}$$

Take the square root of 0.36.

$$x=\frac{-3±\frac{3}{5}}{2\left(-0.2\right)}$$

Multiply 2 times -0.2.

$$x=\frac{-3±\frac{3}{5}}{-0.4}$$

Now solve the equation x=\frac{-3±\frac{3}{5}}{-0.4} when ± is plus. Add -3 to \frac{3}{5}.

$$x=-\frac{\frac{12}{5}}{-0.4}$$

Divide -\frac{12}{5} by -0.4 by multiplying -\frac{12}{5} by the reciprocal of -0.4.

$$x=6$$

Now solve the equation x=\frac{-3±\frac{3}{5}}{-0.4} when ± is minus. Subtract \frac{3}{5} from -3.

$$x=-\frac{\frac{18}{5}}{-0.4}$$

Divide -\frac{18}{5} by -0.4 by multiplying -\frac{18}{5} by the reciprocal of -0.4.

$$x=9$$

The equation is now solved.

$$x=6 x=9$$

Steps for Completing the Square

Use the distributive property to multiply 0.05 by 2x+12.

$$6x=\left(0.1x+0.6\right)\left(2x+18\right)$$

Use the distributive property to multiply 0.1x+0.6 by 2x+18 and combine like terms.

$$6x=0.2x^{2}+3x+10.8$$

Subtract 0.2x^{2} from both sides.

$$6x-0.2x^{2}=3x+10.8$$

Subtract 3x from both sides.

$$6x-0.2x^{2}-3x=10.8$$

Combine 6x and -3x to get 3x.

$$3x-0.2x^{2}=10.8$$

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.

$$-0.2x^{2}+3x=10.8$$

Multiply both sides by -5.

$$\frac{-0.2x^{2}+3x}{-0.2}=\frac{10.8}{-0.2}$$

Dividing by -0.2 undoes the multiplication by -0.2.

$$x^{2}+\frac{3}{-0.2}x=\frac{10.8}{-0.2}$$

Divide 3 by -0.2 by multiplying 3 by the reciprocal of -0.2.

$$x^{2}-15x=\frac{10.8}{-0.2}$$

Divide 10.8 by -0.2 by multiplying 10.8 by the reciprocal of -0.2.

$$x^{2}-15x=-54$$

Divide -15, the coefficient of the x term, by 2 to get -\frac{15}{2}. Then add the square of -\frac{15}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-15x+\left(-\frac{15}{2}\right)^{2}=-54+\left(-\frac{15}{2}\right)^{2}$$

Square -\frac{15}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-15x+\frac{225}{4}=-54+\frac{225}{4}$$

Add -54 to \frac{225}{4}.

$$x^{2}-15x+\frac{225}{4}=\frac{9}{4}$$

Factor x^{2}-15x+\frac{225}{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}.

$$\left(x-\frac{15}{2}\right)^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{15}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Simplify.

$$x-\frac{15}{2}=\frac{3}{2} x-\frac{15}{2}=-\frac{3}{2}$$

Add \frac{15}{2} to both sides of the equation.

$$x=9 x=6$$