Solve left(x-5right)^2=x-7(5+x-7)

$${ \left(x-5 \right) }^{ 2 } =x-7(5+x-7)$$

Solve for x (complex solution)

$$x=2+\sqrt{7}i\approx 2+2.645751311i$$

$$x=-\sqrt{7}i+2\approx 2-2.645751311i$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-5\right)^{2}.

$$x^{2}-10x+25=x-7\left(5+x-7\right)$$

Subtract 7 from 5 to get -2.

$$x^{2}-10x+25=x-7\left(-2+x\right)$$

Use the distributive property to multiply -7 by -2+x.

$$x^{2}-10x+25=x+14-7x$$

Combine x and -7x to get -6x.

$$x^{2}-10x+25=-6x+14$$

Add 6x to both sides.

$$x^{2}-10x+25+6x=14$$

Combine -10x and 6x to get -4x.

$$x^{2}-4x+25=14$$

Subtract 14 from both sides.

$$x^{2}-4x+25-14=0$$

Subtract 14 from 25 to get 11.

$$x^{2}-4x+11=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -4 for b, and 11 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 11}}{2}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 11}}{2}$$

Multiply -4 times 11.

$$x=\frac{-\left(-4\right)±\sqrt{16-44}}{2}$$

Add 16 to -44.

$$x=\frac{-\left(-4\right)±\sqrt{-28}}{2}$$

Take the square root of -28.

$$x=\frac{-\left(-4\right)±2\sqrt{7}i}{2}$$

The opposite of -4 is 4.

$$x=\frac{4±2\sqrt{7}i}{2}$$

Now solve the equation x=\frac{4±2\sqrt{7}i}{2} when ± is plus. Add 4 to 2i\sqrt{7}.

$$x=\frac{4+2\sqrt{7}i}{2}$$

Divide 4+2i\sqrt{7} by 2.

$$x=2+\sqrt{7}i$$

Now solve the equation x=\frac{4±2\sqrt{7}i}{2} when ± is minus. Subtract 2i\sqrt{7} from 4.

$$x=\frac{-2\sqrt{7}i+4}{2}$$

Divide 4-2i\sqrt{7} by 2.

$$x=-\sqrt{7}i+2$$

The equation is now solved.

$$x=2+\sqrt{7}i x=-\sqrt{7}i+2$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-5\right)^{2}.

$$x^{2}-10x+25=x-7\left(5+x-7\right)$$

Subtract 7 from 5 to get -2.

$$x^{2}-10x+25=x-7\left(-2+x\right)$$

Use the distributive property to multiply -7 by -2+x.

$$x^{2}-10x+25=x+14-7x$$

Combine x and -7x to get -6x.

$$x^{2}-10x+25=-6x+14$$

Add 6x to both sides.

$$x^{2}-10x+25+6x=14$$

Combine -10x and 6x to get -4x.

$$x^{2}-4x+25=14$$

Subtract 25 from both sides.

$$x^{2}-4x=14-25$$

Subtract 25 from 14 to get -11.

$$x^{2}-4x=-11$$

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

$$x^{2}-4x+\left(-2\right)^{2}=-11+\left(-2\right)^{2}$$

Square -2.

$$x^{2}-4x+4=-11+4$$

Add -11 to 4.

$$x^{2}-4x+4=-7$$

Factor x^{2}-4x+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-2\right)^{2}=-7$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{-7}$$

Simplify.

$$x-2=\sqrt{7}i x-2=-\sqrt{7}i$$

Add 2 to both sides of the equation.

$$x=2+\sqrt{7}i x=-\sqrt{7}i+2$$