Solve x^2+400-(x-12)^2/40x=x^2+2500-(x-12)^2/100x

$$\frac { x ^ { 2 } + 400 - ( x - 12 ) ^ { 2 } } { 40 x } = \frac { x ^ { 2 } + 2500 - ( x - 12 ) ^ { 2 } } { 100 x }$$

Solve for x

$$x = \frac{143}{3} = 47\frac{2}{3} \approx 47.666666667$$

Steps for Solving Linear Equation

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 200x, the least common multiple of 40x,100x.

$$5\left(x^{2}+400-\left(x-12\right)^{2}\right)=2\left(x^{2}+2500-\left(x-12\right)^{2}\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-12\right)^{2}.

$$5\left(x^{2}+400-\left(x^{2}-24x+144\right)\right)=2\left(x^{2}+2500-\left(x-12\right)^{2}\right)$$

To find the opposite of x^{2}-24x+144, find the opposite of each term.

$$5\left(x^{2}+400-x^{2}+24x-144\right)=2\left(x^{2}+2500-\left(x-12\right)^{2}\right)$$

Combine x^{2} and -x^{2} to get 0.

$$5\left(400+24x-144\right)=2\left(x^{2}+2500-\left(x-12\right)^{2}\right)$$

Subtract 144 from 400 to get 256.

$$5\left(256+24x\right)=2\left(x^{2}+2500-\left(x-12\right)^{2}\right)$$

Use the distributive property to multiply 5 by 256+24x.

$$1280+120x=2\left(x^{2}+2500-\left(x-12\right)^{2}\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-12\right)^{2}.

$$1280+120x=2\left(x^{2}+2500-\left(x^{2}-24x+144\right)\right)$$

To find the opposite of x^{2}-24x+144, find the opposite of each term.

$$1280+120x=2\left(x^{2}+2500-x^{2}+24x-144\right)$$

Combine x^{2} and -x^{2} to get 0.

$$1280+120x=2\left(2500+24x-144\right)$$

Subtract 144 from 2500 to get 2356.

$$1280+120x=2\left(2356+24x\right)$$

Use the distributive property to multiply 2 by 2356+24x.

$$1280+120x=4712+48x$$

Subtract 48x from both sides.

$$1280+120x-48x=4712$$

Combine 120x and -48x to get 72x.

$$1280+72x=4712$$

Subtract 1280 from both sides.

$$72x=4712-1280$$

Subtract 1280 from 4712 to get 3432.

$$72x=3432$$

Divide both sides by 72.

$$x=\frac{3432}{72}$$

Reduce the fraction \frac{3432}{72} to lowest terms by extracting and canceling out 24.

$$x=\frac{143}{3}$$