Solve sqrt{x^2-2x+6}=2x-3

$$\sqrt{ { x }^{ 2 } -2x+6 } = 2x-3$$

Solve for x

$$x=3$$

Solution Steps

Square both sides of the equation.

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

Calculate \sqrt{x^{2}-2x+6} to the power of 2 and get x^{2}-2x+6.

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

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

$$x^{2}-2x+6=4x^{2}-12x+9$$

Subtract 4x^{2} from both sides.

$$x^{2}-2x+6-4x^{2}=-12x+9$$

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

$$-3x^{2}-2x+6=-12x+9$$

Add 12x to both sides.

$$-3x^{2}-2x+6+12x=9$$

Combine -2x and 12x to get 10x.

$$-3x^{2}+10x+6=9$$

Subtract 9 from both sides.

$$-3x^{2}+10x+6-9=0$$

Subtract 9 from 6 to get -3.

$$-3x^{2}+10x-3=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -3x^{2}+ax+bx-3. To find a and b, set up a system to be solved.

$$a+b=10 ab=-3\left(-3\right)=9$$

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 9.

$$1,9 3,3$$

Calculate the sum for each pair.

$$1+9=10 3+3=6$$

The solution is the pair that gives sum 10.

$$a=9 b=1$$

Rewrite -3x^{2}+10x-3 as \left(-3x^{2}+9x\right)+\left(x-3\right).

$$\left(-3x^{2}+9x\right)+\left(x-3\right)$$

Factor out 3x in the first and -1 in the second group.

$$3x\left(-x+3\right)-\left(-x+3\right)$$

Factor out common term -x+3 by using distributive property.

$$\left(-x+3\right)\left(3x-1\right)$$

To find equation solutions, solve -x+3=0 and 3x-1=0.

$$x=3 x=\frac{1}{3}$$

Substitute 3 for x in the equation \sqrt{x^{2}-2x+6}=2x-3.

$$\sqrt{3^{2}-2\times 3+6}=2\times 3-3$$

Simplify. The value x=3 satisfies the equation.

$$3=3$$

Substitute \frac{1}{3} for x in the equation \sqrt{x^{2}-2x+6}=2x-3.

$$\sqrt{\left(\frac{1}{3}\right)^{2}-2\times \frac{1}{3}+6}=2\times \frac{1}{3}-3$$

Simplify. The value x=\frac{1}{3} does not satisfy the equation because the left and the right hand side have opposite signs.

$$\frac{7}{3}=-\frac{7}{3}$$

Equation \sqrt{x^{2}-2x+6}=2x-3 has a unique solution.

$$x=3$$