Solve x^2+(14-x)^2=170

$$x ^ { 2 } + ( 14 - x ) ^ { 2 } = 170$$

Solve for x

$$x=1$$

$$x=13$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(14-x\right)^{2}.

$$x^{2}+196-28x+x^{2}=170$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}+196-28x=170$$

Subtract 170 from both sides.

$$2x^{2}+196-28x-170=0$$

Subtract 170 from 196 to get 26.

$$2x^{2}+26-28x=0$$

Divide both sides by 2.

$$x^{2}+13-14x=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$x^{2}-14x+13=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+13. To find a and b, set up a system to be solved.

$$a+b=-14 ab=1\times 13=13$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. The only such pair is the system solution.

$$a=-13 b=-1$$

Rewrite x^{2}-14x+13 as \left(x^{2}-13x\right)+\left(-x+13\right).

$$\left(x^{2}-13x\right)+\left(-x+13\right)$$

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

$$x\left(x-13\right)-\left(x-13\right)$$

Factor out common term x-13 by using distributive property.

$$\left(x-13\right)\left(x-1\right)$$

To find equation solutions, solve x-13=0 and x-1=0.

$$x=13 x=1$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(14-x\right)^{2}.

$$x^{2}+196-28x+x^{2}=170$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}+196-28x=170$$

Subtract 170 from both sides.

$$2x^{2}+196-28x-170=0$$

Subtract 170 from 196 to get 26.

$$2x^{2}+26-28x=0$$

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.

$$2x^{2}-28x+26=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -28 for b, and 26 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-28\right)±\sqrt{\left(-28\right)^{2}-4\times 2\times 26}}{2\times 2}$$

Square -28.

$$x=\frac{-\left(-28\right)±\sqrt{784-4\times 2\times 26}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-28\right)±\sqrt{784-8\times 26}}{2\times 2}$$

Multiply -8 times 26.

$$x=\frac{-\left(-28\right)±\sqrt{784-208}}{2\times 2}$$

Add 784 to -208.

$$x=\frac{-\left(-28\right)±\sqrt{576}}{2\times 2}$$

Take the square root of 576.

$$x=\frac{-\left(-28\right)±24}{2\times 2}$$

The opposite of -28 is 28.

$$x=\frac{28±24}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{28±24}{4}$$

Now solve the equation x=\frac{28±24}{4} when ± is plus. Add 28 to 24.

$$x=\frac{52}{4}$$

Divide 52 by 4.

$$x=13$$

Now solve the equation x=\frac{28±24}{4} when ± is minus. Subtract 24 from 28.

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

Divide 4 by 4.

$$x=1$$

The equation is now solved.

$$x=13 x=1$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(14-x\right)^{2}.

$$x^{2}+196-28x+x^{2}=170$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}+196-28x=170$$

Subtract 196 from both sides.

$$2x^{2}-28x=170-196$$

Subtract 196 from 170 to get -26.

$$2x^{2}-28x=-26$$

Divide both sides by 2.

$$\frac{2x^{2}-28x}{2}=-\frac{26}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{28}{2}\right)x=-\frac{26}{2}$$

Divide -28 by 2.

$$x^{2}-14x=-\frac{26}{2}$$

Divide -26 by 2.

$$x^{2}-14x=-13$$

Divide -14, the coefficient of the x term, by 2 to get -7. Then add the square of -7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-14x+\left(-7\right)^{2}=-13+\left(-7\right)^{2}$$

Square -7.

$$x^{2}-14x+49=-13+49$$

Add -13 to 49.

$$x^{2}-14x+49=36$$

Factor x^{2}-14x+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-7\right)^{2}=36$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{36}$$

Simplify.

$$x-7=6 x-7=-6$$

Add 7 to both sides of the equation.

$$x=13 x=1$$