Solve (x-6)^2-2x(x+3)=30-12x

$$( x - 6 ) ^ { 2 } - 2 x ( x + 3 ) = 30 - 12 x$$

Solve for x (complex solution)

$$x=\sqrt{15}-3\approx 0.872983346$$

$$x=-\left(\sqrt{15}+3\right)\approx -6.872983346$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-6\right)^{2}.

$$x^{2}-12x+36-2x\left(x+3\right)=30-12x$$

Subtract 30 from both sides.

$$x^{2}-12x+36-2x\left(x+3\right)-30=-12x$$

Add 12x to both sides.

$$x^{2}-12x+36-2x\left(x+3\right)-30+12x=0$$

Use the distributive property to multiply -2x by x+3.

$$x^{2}-12x+36-2x^{2}-6x-30+12x=0$$

Combine x^{2} and -2x^{2} to get -x^{2}.

$$-x^{2}-12x+36-6x-30+12x=0$$

Combine -12x and -6x to get -18x.

$$-x^{2}-18x+36-30+12x=0$$

Subtract 30 from 36 to get 6.

$$-x^{2}-18x+6+12x=0$$

Combine -18x and 12x to get -6x.

$$-x^{2}-6x+6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -6 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-1\right)\times 6}}{2\left(-1\right)}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-1\right)\times 6}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-6\right)±\sqrt{36+4\times 6}}{2\left(-1\right)}$$

Multiply 4 times 6.

$$x=\frac{-\left(-6\right)±\sqrt{36+24}}{2\left(-1\right)}$$

Add 36 to 24.

$$x=\frac{-\left(-6\right)±\sqrt{60}}{2\left(-1\right)}$$

Take the square root of 60.

$$x=\frac{-\left(-6\right)±2\sqrt{15}}{2\left(-1\right)}$$

The opposite of -6 is 6.

$$x=\frac{6±2\sqrt{15}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{6±2\sqrt{15}}{-2}$$

Now solve the equation x=\frac{6±2\sqrt{15}}{-2} when ± is plus. Add 6 to 2\sqrt{15}.

$$x=\frac{2\sqrt{15}+6}{-2}$$

Divide 6+2\sqrt{15} by -2.

$$x=-\left(\sqrt{15}+3\right)$$

Now solve the equation x=\frac{6±2\sqrt{15}}{-2} when ± is minus. Subtract 2\sqrt{15} from 6.

$$x=\frac{6-2\sqrt{15}}{-2}$$

Divide 6-2\sqrt{15} by -2.

$$x=\sqrt{15}-3$$

The equation is now solved.

$$x=-\left(\sqrt{15}+3\right) x=\sqrt{15}-3$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-6\right)^{2}.

$$x^{2}-12x+36-2x\left(x+3\right)=30-12x$$

Add 12x to both sides.

$$x^{2}-12x+36-2x\left(x+3\right)+12x=30$$

Use the distributive property to multiply -2x by x+3.

$$x^{2}-12x+36-2x^{2}-6x+12x=30$$

Combine x^{2} and -2x^{2} to get -x^{2}.

$$-x^{2}-12x+36-6x+12x=30$$

Combine -12x and -6x to get -18x.

$$-x^{2}-18x+36+12x=30$$

Combine -18x and 12x to get -6x.

$$-x^{2}-6x+36=30$$

Subtract 36 from both sides.

$$-x^{2}-6x=30-36$$

Subtract 36 from 30 to get -6.

$$-x^{2}-6x=-6$$

Divide both sides by -1.

$$\frac{-x^{2}-6x}{-1}=-\frac{6}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{6}{-1}\right)x=-\frac{6}{-1}$$

Divide -6 by -1.

$$x^{2}+6x=-\frac{6}{-1}$$

Divide -6 by -1.

$$x^{2}+6x=6$$

Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square 3.

$$x^{2}+6x+9=6+9$$

Add 6 to 9.

$$x^{2}+6x+9=15$$

Factor x^{2}+6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+3\right)^{2}=15$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{15}$$

Simplify.

$$x+3=\sqrt{15} x+3=-\sqrt{15}$$

Subtract 3 from both sides of the equation.

$$x=\sqrt{15}-3 x=-\sqrt{15}-3$$

Solve for x

$$x=\sqrt{15}-3\approx 0.872983346$$

$$x=-\sqrt{15}-3\approx -6.872983346$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-6\right)^{2}.

$$x^{2}-12x+36-2x\left(x+3\right)=30-12x$$

Subtract 30 from both sides.

$$x^{2}-12x+36-2x\left(x+3\right)-30=-12x$$

Add 12x to both sides.

$$x^{2}-12x+36-2x\left(x+3\right)-30+12x=0$$

Use the distributive property to multiply -2x by x+3.

$$x^{2}-12x+36-2x^{2}-6x-30+12x=0$$

Combine x^{2} and -2x^{2} to get -x^{2}.

$$-x^{2}-12x+36-6x-30+12x=0$$

Combine -12x and -6x to get -18x.

$$-x^{2}-18x+36-30+12x=0$$

Subtract 30 from 36 to get 6.

$$-x^{2}-18x+6+12x=0$$

Combine -18x and 12x to get -6x.

$$-x^{2}-6x+6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -6 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-1\right)\times 6}}{2\left(-1\right)}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-1\right)\times 6}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-6\right)±\sqrt{36+4\times 6}}{2\left(-1\right)}$$

Multiply 4 times 6.

$$x=\frac{-\left(-6\right)±\sqrt{36+24}}{2\left(-1\right)}$$

Add 36 to 24.

$$x=\frac{-\left(-6\right)±\sqrt{60}}{2\left(-1\right)}$$

Take the square root of 60.

$$x=\frac{-\left(-6\right)±2\sqrt{15}}{2\left(-1\right)}$$

The opposite of -6 is 6.

$$x=\frac{6±2\sqrt{15}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{6±2\sqrt{15}}{-2}$$

Now solve the equation x=\frac{6±2\sqrt{15}}{-2} when ± is plus. Add 6 to 2\sqrt{15}.

$$x=\frac{2\sqrt{15}+6}{-2}$$

Divide 6+2\sqrt{15} by -2.

$$x=-\left(\sqrt{15}+3\right)$$

Now solve the equation x=\frac{6±2\sqrt{15}}{-2} when ± is minus. Subtract 2\sqrt{15} from 6.

$$x=\frac{6-2\sqrt{15}}{-2}$$

Divide 6-2\sqrt{15} by -2.

$$x=\sqrt{15}-3$$

The equation is now solved.

$$x=-\left(\sqrt{15}+3\right) x=\sqrt{15}-3$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-6\right)^{2}.

$$x^{2}-12x+36-2x\left(x+3\right)=30-12x$$

Add 12x to both sides.

$$x^{2}-12x+36-2x\left(x+3\right)+12x=30$$

Use the distributive property to multiply -2x by x+3.

$$x^{2}-12x+36-2x^{2}-6x+12x=30$$

Combine x^{2} and -2x^{2} to get -x^{2}.

$$-x^{2}-12x+36-6x+12x=30$$

Combine -12x and -6x to get -18x.

$$-x^{2}-18x+36+12x=30$$

Combine -18x and 12x to get -6x.

$$-x^{2}-6x+36=30$$

Subtract 36 from both sides.

$$-x^{2}-6x=30-36$$

Subtract 36 from 30 to get -6.

$$-x^{2}-6x=-6$$

Divide both sides by -1.

$$\frac{-x^{2}-6x}{-1}=-\frac{6}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{6}{-1}\right)x=-\frac{6}{-1}$$

Divide -6 by -1.

$$x^{2}+6x=-\frac{6}{-1}$$

Divide -6 by -1.

$$x^{2}+6x=6$$

Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square 3.

$$x^{2}+6x+9=6+9$$

Add 6 to 9.

$$x^{2}+6x+9=15$$

Factor x^{2}+6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+3\right)^{2}=15$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{15}$$

Simplify.

$$x+3=\sqrt{15} x+3=-\sqrt{15}$$

Subtract 3 from both sides of the equation.

$$x=\sqrt{15}-3 x=-\sqrt{15}-3$$