Solve (x-12)^2+4=2(x-3)^2+1

$$( x - 12 ) ^ { 2 } + 4 = 2 ( x - 3 ) ^ { 2 } + 1$$

Solve for x (complex solution)

$$x=\sqrt{165}-6\approx 6.845232579$$

$$x=-\left(\sqrt{165}+6\right)\approx -18.845232579$$

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}.

$$x^{2}-24x+144+4=2\left(x-3\right)^{2}+1$$

Add 144 and 4 to get 148.

$$x^{2}-24x+148=2\left(x-3\right)^{2}+1$$

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

$$x^{2}-24x+148=2\left(x^{2}-6x+9\right)+1$$

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

$$x^{2}-24x+148=2x^{2}-12x+18+1$$

Add 18 and 1 to get 19.

$$x^{2}-24x+148=2x^{2}-12x+19$$

Subtract 2x^{2} from both sides.

$$x^{2}-24x+148-2x^{2}=-12x+19$$

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

$$-x^{2}-24x+148=-12x+19$$

Add 12x to both sides.

$$-x^{2}-24x+148+12x=19$$

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

$$-x^{2}-12x+148=19$$

Subtract 19 from both sides.

$$-x^{2}-12x+148-19=0$$

Subtract 19 from 148 to get 129.

$$-x^{2}-12x+129=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -12 for b, and 129 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\left(-1\right)\times 129}}{2\left(-1\right)}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\left(-1\right)\times 129}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-12\right)±\sqrt{144+4\times 129}}{2\left(-1\right)}$$

Multiply 4 times 129.

$$x=\frac{-\left(-12\right)±\sqrt{144+516}}{2\left(-1\right)}$$

Add 144 to 516.

$$x=\frac{-\left(-12\right)±\sqrt{660}}{2\left(-1\right)}$$

Take the square root of 660.

$$x=\frac{-\left(-12\right)±2\sqrt{165}}{2\left(-1\right)}$$

The opposite of -12 is 12.

$$x=\frac{12±2\sqrt{165}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{12±2\sqrt{165}}{-2}$$

Now solve the equation x=\frac{12±2\sqrt{165}}{-2} when ± is plus. Add 12 to 2\sqrt{165}.

$$x=\frac{2\sqrt{165}+12}{-2}$$

Divide 12+2\sqrt{165} by -2.

$$x=-\left(\sqrt{165}+6\right)$$

Now solve the equation x=\frac{12±2\sqrt{165}}{-2} when ± is minus. Subtract 2\sqrt{165} from 12.

$$x=\frac{12-2\sqrt{165}}{-2}$$

Divide 12-2\sqrt{165} by -2.

$$x=\sqrt{165}-6$$

The equation is now solved.

$$x=-\left(\sqrt{165}+6\right) x=\sqrt{165}-6$$

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}.

$$x^{2}-24x+144+4=2\left(x-3\right)^{2}+1$$

Add 144 and 4 to get 148.

$$x^{2}-24x+148=2\left(x-3\right)^{2}+1$$

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

$$x^{2}-24x+148=2\left(x^{2}-6x+9\right)+1$$

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

$$x^{2}-24x+148=2x^{2}-12x+18+1$$

Add 18 and 1 to get 19.

$$x^{2}-24x+148=2x^{2}-12x+19$$

Subtract 2x^{2} from both sides.

$$x^{2}-24x+148-2x^{2}=-12x+19$$

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

$$-x^{2}-24x+148=-12x+19$$

Add 12x to both sides.

$$-x^{2}-24x+148+12x=19$$

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

$$-x^{2}-12x+148=19$$

Subtract 148 from both sides.

$$-x^{2}-12x=19-148$$

Subtract 148 from 19 to get -129.

$$-x^{2}-12x=-129$$

Divide both sides by -1.

$$\frac{-x^{2}-12x}{-1}=-\frac{129}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{12}{-1}\right)x=-\frac{129}{-1}$$

Divide -12 by -1.

$$x^{2}+12x=-\frac{129}{-1}$$

Divide -129 by -1.

$$x^{2}+12x=129$$

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.

$$x^{2}+12x+6^{2}=129+6^{2}$$

Square 6.

$$x^{2}+12x+36=129+36$$

Add 129 to 36.

$$x^{2}+12x+36=165$$

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}.

$$\left(x+6\right)^{2}=165$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+6\right)^{2}}=\sqrt{165}$$

Simplify.

$$x+6=\sqrt{165} x+6=-\sqrt{165}$$

Subtract 6 from both sides of the equation.

$$x=\sqrt{165}-6 x=-\sqrt{165}-6$$

Solve for x

$$x=\sqrt{165}-6\approx 6.845232579$$

$$x=-\sqrt{165}-6\approx -18.845232579$$

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}.

$$x^{2}-24x+144+4=2\left(x-3\right)^{2}+1$$

Add 144 and 4 to get 148.

$$x^{2}-24x+148=2\left(x-3\right)^{2}+1$$

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

$$x^{2}-24x+148=2\left(x^{2}-6x+9\right)+1$$

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

$$x^{2}-24x+148=2x^{2}-12x+18+1$$

Add 18 and 1 to get 19.

$$x^{2}-24x+148=2x^{2}-12x+19$$

Subtract 2x^{2} from both sides.

$$x^{2}-24x+148-2x^{2}=-12x+19$$

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

$$-x^{2}-24x+148=-12x+19$$

Add 12x to both sides.

$$-x^{2}-24x+148+12x=19$$

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

$$-x^{2}-12x+148=19$$

Subtract 19 from both sides.

$$-x^{2}-12x+148-19=0$$

Subtract 19 from 148 to get 129.

$$-x^{2}-12x+129=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -12 for b, and 129 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\left(-1\right)\times 129}}{2\left(-1\right)}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\left(-1\right)\times 129}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-12\right)±\sqrt{144+4\times 129}}{2\left(-1\right)}$$

Multiply 4 times 129.

$$x=\frac{-\left(-12\right)±\sqrt{144+516}}{2\left(-1\right)}$$

Add 144 to 516.

$$x=\frac{-\left(-12\right)±\sqrt{660}}{2\left(-1\right)}$$

Take the square root of 660.

$$x=\frac{-\left(-12\right)±2\sqrt{165}}{2\left(-1\right)}$$

The opposite of -12 is 12.

$$x=\frac{12±2\sqrt{165}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{12±2\sqrt{165}}{-2}$$

Now solve the equation x=\frac{12±2\sqrt{165}}{-2} when ± is plus. Add 12 to 2\sqrt{165}.

$$x=\frac{2\sqrt{165}+12}{-2}$$

Divide 12+2\sqrt{165} by -2.

$$x=-\left(\sqrt{165}+6\right)$$

Now solve the equation x=\frac{12±2\sqrt{165}}{-2} when ± is minus. Subtract 2\sqrt{165} from 12.

$$x=\frac{12-2\sqrt{165}}{-2}$$

Divide 12-2\sqrt{165} by -2.

$$x=\sqrt{165}-6$$

The equation is now solved.

$$x=-\left(\sqrt{165}+6\right) x=\sqrt{165}-6$$

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}.

$$x^{2}-24x+144+4=2\left(x-3\right)^{2}+1$$

Add 144 and 4 to get 148.

$$x^{2}-24x+148=2\left(x-3\right)^{2}+1$$

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

$$x^{2}-24x+148=2\left(x^{2}-6x+9\right)+1$$

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

$$x^{2}-24x+148=2x^{2}-12x+18+1$$

Add 18 and 1 to get 19.

$$x^{2}-24x+148=2x^{2}-12x+19$$

Subtract 2x^{2} from both sides.

$$x^{2}-24x+148-2x^{2}=-12x+19$$

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

$$-x^{2}-24x+148=-12x+19$$

Add 12x to both sides.

$$-x^{2}-24x+148+12x=19$$

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

$$-x^{2}-12x+148=19$$

Subtract 148 from both sides.

$$-x^{2}-12x=19-148$$

Subtract 148 from 19 to get -129.

$$-x^{2}-12x=-129$$

Divide both sides by -1.

$$\frac{-x^{2}-12x}{-1}=-\frac{129}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{12}{-1}\right)x=-\frac{129}{-1}$$

Divide -12 by -1.

$$x^{2}+12x=-\frac{129}{-1}$$

Divide -129 by -1.

$$x^{2}+12x=129$$

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.

$$x^{2}+12x+6^{2}=129+6^{2}$$

Square 6.

$$x^{2}+12x+36=129+36$$

Add 129 to 36.

$$x^{2}+12x+36=165$$

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}.

$$\left(x+6\right)^{2}=165$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+6\right)^{2}}=\sqrt{165}$$

Simplify.

$$x+6=\sqrt{165} x+6=-\sqrt{165}$$

Subtract 6 from both sides of the equation.

$$x=\sqrt{165}-6 x=-\sqrt{165}-6$$