Solve (2x-3)^2=(3x-1)(x-1)=

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

Solve for x

$$x=2\sqrt{2}+4\approx 6.828427125$$

$$x=4-2\sqrt{2}\approx 1.171572875$$

Steps Using the Quadratic Formula

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

$$4x^{2}-12x+9=\left(3x-1\right)\left(x-1\right)$$

Use the distributive property to multiply 3x-1 by x-1 and combine like terms.

$$4x^{2}-12x+9=3x^{2}-4x+1$$

Subtract 3x^{2} from both sides.

$$4x^{2}-12x+9-3x^{2}=-4x+1$$

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

$$x^{2}-12x+9=-4x+1$$

Add 4x to both sides.

$$x^{2}-12x+9+4x=1$$

Combine -12x and 4x to get -8x.

$$x^{2}-8x+9=1$$

Subtract 1 from both sides.

$$x^{2}-8x+9-1=0$$

Subtract 1 from 9 to get 8.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -8 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 8}}{2}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 8}}{2}$$

Multiply -4 times 8.

$$x=\frac{-\left(-8\right)±\sqrt{64-32}}{2}$$

Add 64 to -32.

$$x=\frac{-\left(-8\right)±\sqrt{32}}{2}$$

Take the square root of 32.

$$x=\frac{-\left(-8\right)±4\sqrt{2}}{2}$$

The opposite of -8 is 8.

$$x=\frac{8±4\sqrt{2}}{2}$$

Now solve the equation x=\frac{8±4\sqrt{2}}{2} when ± is plus. Add 8 to 4\sqrt{2}.

$$x=\frac{4\sqrt{2}+8}{2}$$

Divide 8+4\sqrt{2} by 2.

$$x=2\sqrt{2}+4$$

Now solve the equation x=\frac{8±4\sqrt{2}}{2} when ± is minus. Subtract 4\sqrt{2} from 8.

$$x=\frac{8-4\sqrt{2}}{2}$$

Divide 8-4\sqrt{2} by 2.

$$x=4-2\sqrt{2}$$

The equation is now solved.

$$x=2\sqrt{2}+4 x=4-2\sqrt{2}$$

Steps for Completing the Square

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

$$4x^{2}-12x+9=\left(3x-1\right)\left(x-1\right)$$

Use the distributive property to multiply 3x-1 by x-1 and combine like terms.

$$4x^{2}-12x+9=3x^{2}-4x+1$$

Subtract 3x^{2} from both sides.

$$4x^{2}-12x+9-3x^{2}=-4x+1$$

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

$$x^{2}-12x+9=-4x+1$$

Add 4x to both sides.

$$x^{2}-12x+9+4x=1$$

Combine -12x and 4x to get -8x.

$$x^{2}-8x+9=1$$

Subtract 9 from both sides.

$$x^{2}-8x=1-9$$

Subtract 9 from 1 to get -8.

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

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.

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

Square -4.

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

Add -8 to 16.

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

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-4=2\sqrt{2} x-4=-2\sqrt{2}$$

Add 4 to both sides of the equation.

$$x=2\sqrt{2}+4 x=4-2\sqrt{2}$$