Solve 17=1+(x-1)*(x-1)

$$17=1+(x-1) \times (x-1)$$

Solve for x

$$x=5$$

$$x=-3$$

Steps Using the Quadratic Formula

Multiply x-1 and x-1 to get \left(x-1\right)^{2}.

$$17=1+\left(x-1\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$17=1+x^{2}-2x+1$$

Add 1 and 1 to get 2.

$$17=2+x^{2}-2x$$

Swap sides so that all variable terms are on the left hand side.

$$2+x^{2}-2x=17$$

Subtract 17 from both sides.

$$2+x^{2}-2x-17=0$$

Subtract 17 from 2 to get -15.

$$-15+x^{2}-2x=0$$

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.

$$x^{2}-2x-15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2 for b, and -15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-15\right)}}{2}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\left(-15\right)}}{2}$$

Multiply -4 times -15.

$$x=\frac{-\left(-2\right)±\sqrt{4+60}}{2}$$

Add 4 to 60.

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

Take the square root of 64.

$$x=\frac{-\left(-2\right)±8}{2}$$

The opposite of -2 is 2.

$$x=\frac{2±8}{2}$$

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

$$x=\frac{10}{2}$$

Divide 10 by 2.

$$x=5$$

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

$$x=-\frac{6}{2}$$

Divide -6 by 2.

$$x=-3$$

The equation is now solved.

$$x=5 x=-3$$

Steps for Completing the Square

Multiply x-1 and x-1 to get \left(x-1\right)^{2}.

$$17=1+\left(x-1\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$17=1+x^{2}-2x+1$$

Add 1 and 1 to get 2.

$$17=2+x^{2}-2x$$

Swap sides so that all variable terms are on the left hand side.

$$2+x^{2}-2x=17$$

Subtract 2 from both sides.

$$x^{2}-2x=17-2$$

Subtract 2 from 17 to get 15.

$$x^{2}-2x=15$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=15+1$$

Add 15 to 1.

$$x^{2}-2x+1=16$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=16$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{16}$$

Simplify.

$$x-1=4 x-1=-4$$

Add 1 to both sides of the equation.

$$x=5 x=-3$$