Solve 6^2+(13-x)^2=2x^2

$$6 ^ { 2 } + ( 13 - x ) ^ { 2 } = 2 x ^ { 2 }$$

Solve for x (complex solution)

$$x=\sqrt{374}-13\approx 6.339079606$$

$$x=-\left(\sqrt{374}+13\right)\approx -32.339079606$$

Steps Using the Quadratic Formula

Calculate 6 to the power of 2 and get 36.

$$36+\left(13-x\right)^{2}=2x^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(13-x\right)^{2}.

$$36+169-26x+x^{2}=2x^{2}$$

Add 36 and 169 to get 205.

$$205-26x+x^{2}=2x^{2}$$

Subtract 2x^{2} from both sides.

$$205-26x+x^{2}-2x^{2}=0$$

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

$$205-26x-x^{2}=0$$

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.

$$-x^{2}-26x+205=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -26 for b, and 205 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-26\right)±\sqrt{\left(-26\right)^{2}-4\left(-1\right)\times 205}}{2\left(-1\right)}$$

Square -26.

$$x=\frac{-\left(-26\right)±\sqrt{676-4\left(-1\right)\times 205}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-26\right)±\sqrt{676+4\times 205}}{2\left(-1\right)}$$

Multiply 4 times 205.

$$x=\frac{-\left(-26\right)±\sqrt{676+820}}{2\left(-1\right)}$$

Add 676 to 820.

$$x=\frac{-\left(-26\right)±\sqrt{1496}}{2\left(-1\right)}$$

Take the square root of 1496.

$$x=\frac{-\left(-26\right)±2\sqrt{374}}{2\left(-1\right)}$$

The opposite of -26 is 26.

$$x=\frac{26±2\sqrt{374}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{26±2\sqrt{374}}{-2}$$

Now solve the equation x=\frac{26±2\sqrt{374}}{-2} when ± is plus. Add 26 to 2\sqrt{374}.

$$x=\frac{2\sqrt{374}+26}{-2}$$

Divide 26+2\sqrt{374} by -2.

$$x=-\left(\sqrt{374}+13\right)$$

Now solve the equation x=\frac{26±2\sqrt{374}}{-2} when ± is minus. Subtract 2\sqrt{374} from 26.

$$x=\frac{26-2\sqrt{374}}{-2}$$

Divide 26-2\sqrt{374} by -2.

$$x=\sqrt{374}-13$$

The equation is now solved.

$$x=-\left(\sqrt{374}+13\right) x=\sqrt{374}-13$$

Steps for Completing the Square

Calculate 6 to the power of 2 and get 36.

$$36+\left(13-x\right)^{2}=2x^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(13-x\right)^{2}.

$$36+169-26x+x^{2}=2x^{2}$$

Add 36 and 169 to get 205.

$$205-26x+x^{2}=2x^{2}$$

Subtract 2x^{2} from both sides.

$$205-26x+x^{2}-2x^{2}=0$$

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

$$205-26x-x^{2}=0$$

Subtract 205 from both sides. Anything subtracted from zero gives its negation.

$$-26x-x^{2}=-205$$

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.

$$-x^{2}-26x=-205$$

Divide both sides by -1.

$$\frac{-x^{2}-26x}{-1}=-\frac{205}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{26}{-1}\right)x=-\frac{205}{-1}$$

Divide -26 by -1.

$$x^{2}+26x=-\frac{205}{-1}$$

Divide -205 by -1.

$$x^{2}+26x=205$$

Divide 26, the coefficient of the x term, by 2 to get 13. Then add the square of 13 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+26x+13^{2}=205+13^{2}$$

Square 13.

$$x^{2}+26x+169=205+169$$

Add 205 to 169.

$$x^{2}+26x+169=374$$

Factor x^{2}+26x+169. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+13\right)^{2}=374$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+13\right)^{2}}=\sqrt{374}$$

Simplify.

$$x+13=\sqrt{374} x+13=-\sqrt{374}$$

Subtract 13 from both sides of the equation.

$$x=\sqrt{374}-13 x=-\sqrt{374}-13$$

Solve for x

$$x=\sqrt{374}-13\approx 6.339079606$$

$$x=-\sqrt{374}-13\approx -32.339079606$$

Steps Using the Quadratic Formula

Calculate 6 to the power of 2 and get 36.

$$36+\left(13-x\right)^{2}=2x^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(13-x\right)^{2}.

$$36+169-26x+x^{2}=2x^{2}$$

Add 36 and 169 to get 205.

$$205-26x+x^{2}=2x^{2}$$

Subtract 2x^{2} from both sides.

$$205-26x+x^{2}-2x^{2}=0$$

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

$$205-26x-x^{2}=0$$

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.

$$-x^{2}-26x+205=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -26 for b, and 205 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-26\right)±\sqrt{\left(-26\right)^{2}-4\left(-1\right)\times 205}}{2\left(-1\right)}$$

Square -26.

$$x=\frac{-\left(-26\right)±\sqrt{676-4\left(-1\right)\times 205}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-26\right)±\sqrt{676+4\times 205}}{2\left(-1\right)}$$

Multiply 4 times 205.

$$x=\frac{-\left(-26\right)±\sqrt{676+820}}{2\left(-1\right)}$$

Add 676 to 820.

$$x=\frac{-\left(-26\right)±\sqrt{1496}}{2\left(-1\right)}$$

Take the square root of 1496.

$$x=\frac{-\left(-26\right)±2\sqrt{374}}{2\left(-1\right)}$$

The opposite of -26 is 26.

$$x=\frac{26±2\sqrt{374}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{26±2\sqrt{374}}{-2}$$

Now solve the equation x=\frac{26±2\sqrt{374}}{-2} when ± is plus. Add 26 to 2\sqrt{374}.

$$x=\frac{2\sqrt{374}+26}{-2}$$

Divide 26+2\sqrt{374} by -2.

$$x=-\left(\sqrt{374}+13\right)$$

Now solve the equation x=\frac{26±2\sqrt{374}}{-2} when ± is minus. Subtract 2\sqrt{374} from 26.

$$x=\frac{26-2\sqrt{374}}{-2}$$

Divide 26-2\sqrt{374} by -2.

$$x=\sqrt{374}-13$$

The equation is now solved.

$$x=-\left(\sqrt{374}+13\right) x=\sqrt{374}-13$$

Steps for Completing the Square

Calculate 6 to the power of 2 and get 36.

$$36+\left(13-x\right)^{2}=2x^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(13-x\right)^{2}.

$$36+169-26x+x^{2}=2x^{2}$$

Add 36 and 169 to get 205.

$$205-26x+x^{2}=2x^{2}$$

Subtract 2x^{2} from both sides.

$$205-26x+x^{2}-2x^{2}=0$$

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

$$205-26x-x^{2}=0$$

Subtract 205 from both sides. Anything subtracted from zero gives its negation.

$$-26x-x^{2}=-205$$

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.

$$-x^{2}-26x=-205$$

Divide both sides by -1.

$$\frac{-x^{2}-26x}{-1}=-\frac{205}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{26}{-1}\right)x=-\frac{205}{-1}$$

Divide -26 by -1.

$$x^{2}+26x=-\frac{205}{-1}$$

Divide -205 by -1.

$$x^{2}+26x=205$$

Divide 26, the coefficient of the x term, by 2 to get 13. Then add the square of 13 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+26x+13^{2}=205+13^{2}$$

Square 13.

$$x^{2}+26x+169=205+169$$

Add 205 to 169.

$$x^{2}+26x+169=374$$

Factor x^{2}+26x+169. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+13\right)^{2}=374$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+13\right)^{2}}=\sqrt{374}$$

Simplify.

$$x+13=\sqrt{374} x+13=-\sqrt{374}$$

Subtract 13 from both sides of the equation.

$$x=\sqrt{374}-13 x=-\sqrt{374}-13$$