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

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

Solve for x

$$x=2$$

$$x=4$$

Steps Using Factoring

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

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

Add 4 and 1 to get 5.

$$x^{2}-4x+5=2x-3$$

Subtract 2x from both sides.

$$x^{2}-4x+5-2x=-3$$

Combine -4x and -2x to get -6x.

$$x^{2}-6x+5=-3$$

Add 3 to both sides.

$$x^{2}-6x+5+3=0$$

Add 5 and 3 to get 8.

$$x^{2}-6x+8=0$$

To solve the equation, factor x^{2}-6x+8 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-6 ab=8$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 8.

$$-1,-8 -2,-4$$

Calculate the sum for each pair.

$$-1-8=-9 -2-4=-6$$

The solution is the pair that gives sum -6.

$$a=-4 b=-2$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-4\right)\left(x-2\right)$$

To find equation solutions, solve x-4=0 and x-2=0.

$$x=4 x=2$$

Steps Using Factoring By Grouping

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

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

Add 4 and 1 to get 5.

$$x^{2}-4x+5=2x-3$$

Subtract 2x from both sides.

$$x^{2}-4x+5-2x=-3$$

Combine -4x and -2x to get -6x.

$$x^{2}-6x+5=-3$$

Add 3 to both sides.

$$x^{2}-6x+5+3=0$$

Add 5 and 3 to get 8.

$$x^{2}-6x+8=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+8. To find a and b, set up a system to be solved.

$$a+b=-6 ab=1\times 8=8$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 8.

$$-1,-8 -2,-4$$

Calculate the sum for each pair.

$$-1-8=-9 -2-4=-6$$

The solution is the pair that gives sum -6.

$$a=-4 b=-2$$

Rewrite x^{2}-6x+8 as \left(x^{2}-4x\right)+\left(-2x+8\right).

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

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

$$x\left(x-4\right)-2\left(x-4\right)$$

Factor out common term x-4 by using distributive property.

$$\left(x-4\right)\left(x-2\right)$$

To find equation solutions, solve x-4=0 and x-2=0.

$$x=4 x=2$$

Steps Using the Quadratic Formula

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

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

Add 4 and 1 to get 5.

$$x^{2}-4x+5=2x-3$$

Subtract 2x from both sides.

$$x^{2}-4x+5-2x=-3$$

Combine -4x and -2x to get -6x.

$$x^{2}-6x+5=-3$$

Add 3 to both sides.

$$x^{2}-6x+5+3=0$$

Add 5 and 3 to get 8.

$$x^{2}-6x+8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -6 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 8}}{2}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 8}}{2}$$

Multiply -4 times 8.

$$x=\frac{-\left(-6\right)±\sqrt{36-32}}{2}$$

Add 36 to -32.

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

Take the square root of 4.

$$x=\frac{-\left(-6\right)±2}{2}$$

The opposite of -6 is 6.

$$x=\frac{6±2}{2}$$

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

$$x=\frac{8}{2}$$

Divide 8 by 2.

$$x=4$$

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

$$x=\frac{4}{2}$$

Divide 4 by 2.

$$x=2$$

The equation is now solved.

$$x=4 x=2$$

Steps for Completing the Square

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

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

Add 4 and 1 to get 5.

$$x^{2}-4x+5=2x-3$$

Subtract 2x from both sides.

$$x^{2}-4x+5-2x=-3$$

Combine -4x and -2x to get -6x.

$$x^{2}-6x+5=-3$$

Subtract 5 from both sides.

$$x^{2}-6x=-3-5$$

Subtract 5 from -3 to get -8.

$$x^{2}-6x=-8$$

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.

$$x^{2}-6x+\left(-3\right)^{2}=-8+\left(-3\right)^{2}$$

Square -3.

$$x^{2}-6x+9=-8+9$$

Add -8 to 9.

$$x^{2}-6x+9=1$$

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}.

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

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{1}$$

Simplify.

$$x-3=1 x-3=-1$$

Add 3 to both sides of the equation.

$$x=4 x=2$$