Solve (x-12)^2+4=2(x-3)^2+2

$$( x - 12 ) ^ { 2 } + 4 = 2 ( x - 3 ) ^ { 2 } + 2$$

Solve for x

$$x=2\sqrt{41}-6\approx 6.806248475$$

$$x=-2\sqrt{41}-6\approx -18.806248475$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-12\right)^{2}.

$$x^{2}-24x+144+4=2\left(x-3\right)^{2}+2$$

Add 144 and 4 to get 148.

$$x^{2}-24x+148=2\left(x-3\right)^{2}+2$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$x^{2}-24x+148=2\left(x^{2}-6x+9\right)+2$$

Use the distributive property to multiply 2 by x^{2}-6x+9.

$$x^{2}-24x+148=2x^{2}-12x+18+2$$

Add 18 and 2 to get 20.

$$x^{2}-24x+148=2x^{2}-12x+20$$

Subtract 2x^{2} from both sides.

$$x^{2}-24x+148-2x^{2}=-12x+20$$

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

$$-x^{2}-24x+148=-12x+20$$

Add 12x to both sides.

$$-x^{2}-24x+148+12x=20$$

Combine -24x and 12x to get -12x.

$$-x^{2}-12x+148=20$$

Subtract 20 from both sides.

$$-x^{2}-12x+148-20=0$$

Subtract 20 from 148 to get 128.

$$-x^{2}-12x+128=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -12 for b, and 128 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\left(-1\right)\times 128}}{2\left(-1\right)}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\left(-1\right)\times 128}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-12\right)±\sqrt{144+4\times 128}}{2\left(-1\right)}$$

Multiply 4 times 128.

$$x=\frac{-\left(-12\right)±\sqrt{144+512}}{2\left(-1\right)}$$

Add 144 to 512.

$$x=\frac{-\left(-12\right)±\sqrt{656}}{2\left(-1\right)}$$

Take the square root of 656.

$$x=\frac{-\left(-12\right)±4\sqrt{41}}{2\left(-1\right)}$$

The opposite of -12 is 12.

$$x=\frac{12±4\sqrt{41}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{12±4\sqrt{41}}{-2}$$

Now solve the equation x=\frac{12±4\sqrt{41}}{-2} when ± is plus. Add 12 to 4\sqrt{41}.

$$x=\frac{4\sqrt{41}+12}{-2}$$

Divide 12+4\sqrt{41} by -2.

$$x=-2\sqrt{41}-6$$

Now solve the equation x=\frac{12±4\sqrt{41}}{-2} when ± is minus. Subtract 4\sqrt{41} from 12.

$$x=\frac{12-4\sqrt{41}}{-2}$$

Divide 12-4\sqrt{41} by -2.

$$x=2\sqrt{41}-6$$

The equation is now solved.

$$x=-2\sqrt{41}-6 x=2\sqrt{41}-6$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-12\right)^{2}.

$$x^{2}-24x+144+4=2\left(x-3\right)^{2}+2$$

Add 144 and 4 to get 148.

$$x^{2}-24x+148=2\left(x-3\right)^{2}+2$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$x^{2}-24x+148=2\left(x^{2}-6x+9\right)+2$$

Use the distributive property to multiply 2 by x^{2}-6x+9.

$$x^{2}-24x+148=2x^{2}-12x+18+2$$

Add 18 and 2 to get 20.

$$x^{2}-24x+148=2x^{2}-12x+20$$

Subtract 2x^{2} from both sides.

$$x^{2}-24x+148-2x^{2}=-12x+20$$

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

$$-x^{2}-24x+148=-12x+20$$

Add 12x to both sides.

$$-x^{2}-24x+148+12x=20$$

Combine -24x and 12x to get -12x.

$$-x^{2}-12x+148=20$$

Subtract 148 from both sides.

$$-x^{2}-12x=20-148$$

Subtract 148 from 20 to get -128.

$$-x^{2}-12x=-128$$

Divide both sides by -1.

$$\frac{-x^{2}-12x}{-1}=-\frac{128}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{12}{-1}\right)x=-\frac{128}{-1}$$

Divide -12 by -1.

$$x^{2}+12x=-\frac{128}{-1}$$

Divide -128 by -1.

$$x^{2}+12x=128$$

Divide 12, the coefficient of the x term, by 2 to get 6. Then add the square of 6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+12x+6^{2}=128+6^{2}$$

Square 6.

$$x^{2}+12x+36=128+36$$

Add 128 to 36.

$$x^{2}+12x+36=164$$

Factor x^{2}+12x+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+6\right)^{2}=164$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+6\right)^{2}}=\sqrt{164}$$

Simplify.

$$x+6=2\sqrt{41} x+6=-2\sqrt{41}$$

Subtract 6 from both sides of the equation.

$$x=2\sqrt{41}-6 x=-2\sqrt{41}-6$$