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

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

Solve for x

$$x=6$$

$$x=2$$

Steps Using Factoring By Grouping

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

$$x^{2}=4\left(x^{2}-6x+9\right)$$

Use the distributive property to multiply 4 by x^{2}-6x+9.

$$x^{2}=4x^{2}-24x+36$$

Subtract 4x^{2} from both sides.

$$x^{2}-4x^{2}=-24x+36$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$-3x^{2}=-24x+36$$

Add 24x to both sides.

$$-3x^{2}+24x=36$$

Subtract 36 from both sides.

$$-3x^{2}+24x-36=0$$

Divide both sides by 3.

$$-x^{2}+8x-12=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-12. To find a and b, set up a system to be solved.

$$a+b=8 ab=-\left(-12\right)=12$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 12.

$$1,12 2,6 3,4$$

Calculate the sum for each pair.

$$1+12=13 2+6=8 3+4=7$$

The solution is the pair that gives sum 8.

$$a=6 b=2$$

Rewrite -x^{2}+8x-12 as \left(-x^{2}+6x\right)+\left(2x-12\right).

$$\left(-x^{2}+6x\right)+\left(2x-12\right)$$

Factor out -x in the first and 2 in the second group.

$$-x\left(x-6\right)+2\left(x-6\right)$$

Factor out common term x-6 by using distributive property.

$$\left(x-6\right)\left(-x+2\right)$$

To find equation solutions, solve x-6=0 and -x+2=0.

$$x=6 x=2$$

Steps Using the Quadratic Formula

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

$$x^{2}=4\left(x^{2}-6x+9\right)$$

Use the distributive property to multiply 4 by x^{2}-6x+9.

$$x^{2}=4x^{2}-24x+36$$

Subtract 4x^{2} from both sides.

$$x^{2}-4x^{2}=-24x+36$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$-3x^{2}=-24x+36$$

Add 24x to both sides.

$$-3x^{2}+24x=36$$

Subtract 36 from both sides.

$$-3x^{2}+24x-36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 24 for b, and -36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-24±\sqrt{24^{2}-4\left(-3\right)\left(-36\right)}}{2\left(-3\right)}$$

Square 24.

$$x=\frac{-24±\sqrt{576-4\left(-3\right)\left(-36\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-24±\sqrt{576+12\left(-36\right)}}{2\left(-3\right)}$$

Multiply 12 times -36.

$$x=\frac{-24±\sqrt{576-432}}{2\left(-3\right)}$$

Add 576 to -432.

$$x=\frac{-24±\sqrt{144}}{2\left(-3\right)}$$

Take the square root of 144.

$$x=\frac{-24±12}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-24±12}{-6}$$

Now solve the equation x=\frac{-24±12}{-6} when ± is plus. Add -24 to 12.

$$x=-\frac{12}{-6}$$

Divide -12 by -6.

$$x=2$$

Now solve the equation x=\frac{-24±12}{-6} when ± is minus. Subtract 12 from -24.

$$x=-\frac{36}{-6}$$

Divide -36 by -6.

$$x=6$$

The equation is now solved.

$$x=2 x=6$$

Steps for Completing the Square

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

$$x^{2}=4\left(x^{2}-6x+9\right)$$

Use the distributive property to multiply 4 by x^{2}-6x+9.

$$x^{2}=4x^{2}-24x+36$$

Subtract 4x^{2} from both sides.

$$x^{2}-4x^{2}=-24x+36$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$-3x^{2}=-24x+36$$

Add 24x to both sides.

$$-3x^{2}+24x=36$$

Divide both sides by -3.

$$\frac{-3x^{2}+24x}{-3}=\frac{36}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{24}{-3}x=\frac{36}{-3}$$

Divide 24 by -3.

$$x^{2}-8x=\frac{36}{-3}$$

Divide 36 by -3.

$$x^{2}-8x=-12$$

Divide -8, the coefficient of the x term, by 2 to get -4. Then add the square of -4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-8x+\left(-4\right)^{2}=-12+\left(-4\right)^{2}$$

Square -4.

$$x^{2}-8x+16=-12+16$$

Add -12 to 16.

$$x^{2}-8x+16=4$$

Factor x^{2}-8x+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-4=2 x-4=-2$$

Add 4 to both sides of the equation.

$$x=6 x=2$$