Solve 2(17-x)=(23-x)^2

$$2 ( 17 - x ) = ( 23 - x ) ^ { 2 }$$

Solve for x (complex solution)

$$x=22+\sqrt{11}i\approx 22+3.31662479i$$

$$x=-\sqrt{11}i+22\approx 22-3.31662479i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2 by 17-x.

$$34-2x=\left(23-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(23-x\right)^{2}.

$$34-2x=529-46x+x^{2}$$

Subtract 529 from both sides.

$$34-2x-529=-46x+x^{2}$$

Subtract 529 from 34 to get -495.

$$-495-2x=-46x+x^{2}$$

Add 46x to both sides.

$$-495-2x+46x=x^{2}$$

Combine -2x and 46x to get 44x.

$$-495+44x=x^{2}$$

Subtract x^{2} from both sides.

$$-495+44x-x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-x^{2}+44x-495=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 44 for b, and -495 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-44±\sqrt{44^{2}-4\left(-1\right)\left(-495\right)}}{2\left(-1\right)}$$

Square 44.

$$x=\frac{-44±\sqrt{1936-4\left(-1\right)\left(-495\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-44±\sqrt{1936+4\left(-495\right)}}{2\left(-1\right)}$$

Multiply 4 times -495.

$$x=\frac{-44±\sqrt{1936-1980}}{2\left(-1\right)}$$

Add 1936 to -1980.

$$x=\frac{-44±\sqrt{-44}}{2\left(-1\right)}$$

Take the square root of -44.

$$x=\frac{-44±2\sqrt{11}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-44±2\sqrt{11}i}{-2}$$

Now solve the equation x=\frac{-44±2\sqrt{11}i}{-2} when ± is plus. Add -44 to 2i\sqrt{11}.

$$x=\frac{-44+2\sqrt{11}i}{-2}$$

Divide -44+2i\sqrt{11} by -2.

$$x=-\sqrt{11}i+22$$

Now solve the equation x=\frac{-44±2\sqrt{11}i}{-2} when ± is minus. Subtract 2i\sqrt{11} from -44.

$$x=\frac{-2\sqrt{11}i-44}{-2}$$

Divide -44-2i\sqrt{11} by -2.

$$x=22+\sqrt{11}i$$

The equation is now solved.

$$x=-\sqrt{11}i+22 x=22+\sqrt{11}i$$

Steps for Completing the Square

Use the distributive property to multiply 2 by 17-x.

$$34-2x=\left(23-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(23-x\right)^{2}.

$$34-2x=529-46x+x^{2}$$

Add 46x to both sides.

$$34-2x+46x=529+x^{2}$$

Combine -2x and 46x to get 44x.

$$34+44x=529+x^{2}$$

Subtract x^{2} from both sides.

$$34+44x-x^{2}=529$$

Subtract 34 from both sides.

$$44x-x^{2}=529-34$$

Subtract 34 from 529 to get 495.

$$44x-x^{2}=495$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-x^{2}+44x=495$$

Divide both sides by -1.

$$\frac{-x^{2}+44x}{-1}=\frac{495}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{44}{-1}x=\frac{495}{-1}$$

Divide 44 by -1.

$$x^{2}-44x=\frac{495}{-1}$$

Divide 495 by -1.

$$x^{2}-44x=-495$$

Divide -44, the coefficient of the x term, by 2 to get -22. Then add the square of -22 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-44x+\left(-22\right)^{2}=-495+\left(-22\right)^{2}$$

Square -22.

$$x^{2}-44x+484=-495+484$$

Add -495 to 484.

$$x^{2}-44x+484=-11$$

Factor x^{2}-44x+484. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-22\right)^{2}=-11$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-22\right)^{2}}=\sqrt{-11}$$

Simplify.

$$x-22=\sqrt{11}i x-22=-\sqrt{11}i$$

Add 22 to both sides of the equation.

$$x=22+\sqrt{11}i x=-\sqrt{11}i+22$$