Solve 24^2+x^2=2(13^2+19^2)

$$24 ^ { 2 } + x ^ { 2 } = 2 ( 13 ^ { 2 } + 19 ^ { 2 } )$$

Solve for x

$$x=-22$$

$$x=22$$

Steps Using Difference of Squares

Calculate 24 to the power of 2 and get 576.

$$576+x^{2}=2\left(13^{2}+19^{2}\right)$$

Calculate 13 to the power of 2 and get 169.

$$576+x^{2}=2\left(169+19^{2}\right)$$

Calculate 19 to the power of 2 and get 361.

$$576+x^{2}=2\left(169+361\right)$$

Add 169 and 361 to get 530.

$$576+x^{2}=2\times 530$$

Multiply 2 and 530 to get 1060.

$$576+x^{2}=1060$$

Subtract 1060 from both sides.

$$576+x^{2}-1060=0$$

Subtract 1060 from 576 to get -484.

$$-484+x^{2}=0$$

Consider -484+x^{2}. Rewrite -484+x^{2} as x^{2}-22^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x-22\right)\left(x+22\right)=0$$

To find equation solutions, solve x-22=0 and x+22=0.

$$x=22 x=-22$$

Steps by Finding Square Root

Calculate 24 to the power of 2 and get 576.

$$576+x^{2}=2\left(13^{2}+19^{2}\right)$$

Calculate 13 to the power of 2 and get 169.

$$576+x^{2}=2\left(169+19^{2}\right)$$

Calculate 19 to the power of 2 and get 361.

$$576+x^{2}=2\left(169+361\right)$$

Add 169 and 361 to get 530.

$$576+x^{2}=2\times 530$$

Multiply 2 and 530 to get 1060.

$$576+x^{2}=1060$$

Subtract 576 from both sides.

$$x^{2}=1060-576$$

Subtract 576 from 1060 to get 484.

$$x^{2}=484$$

Take the square root of both sides of the equation.

$$x=22 x=-22$$

Steps Using the Quadratic Formula

Calculate 24 to the power of 2 and get 576.

$$576+x^{2}=2\left(13^{2}+19^{2}\right)$$

Calculate 13 to the power of 2 and get 169.

$$576+x^{2}=2\left(169+19^{2}\right)$$

Calculate 19 to the power of 2 and get 361.

$$576+x^{2}=2\left(169+361\right)$$

Add 169 and 361 to get 530.

$$576+x^{2}=2\times 530$$

Multiply 2 and 530 to get 1060.

$$576+x^{2}=1060$$

Subtract 1060 from both sides.

$$576+x^{2}-1060=0$$

Subtract 1060 from 576 to get -484.

$$-484+x^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$x^{2}-484=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and -484 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-484\right)}}{2}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-484\right)}}{2}$$

Multiply -4 times -484.

$$x=\frac{0±\sqrt{1936}}{2}$$

Take the square root of 1936.

$$x=\frac{0±44}{2}$$

Now solve the equation x=\frac{0±44}{2} when ± is plus. Divide 44 by 2.

$$x=22$$

Now solve the equation x=\frac{0±44}{2} when ± is minus. Divide -44 by 2.

$$x=-22$$

The equation is now solved.

$$x=22 x=-22$$