Solve 144/25x^2+x^2=52^2

$$\frac { 144 } { 25 } x ^ { 2 } + x ^ { 2 } = 52 ^ { 2 }$$

Solve for x

$$x=20$$

$$x=-20$$

Steps Using Difference of Squares

Combine \frac{144}{25}x^{2} and x^{2} to get \frac{169}{25}x^{2}.

$$\frac{169}{25}x^{2}=52^{2}$$

Calculate 52 to the power of 2 and get 2704.

$$\frac{169}{25}x^{2}=2704$$

Subtract 2704 from both sides.

$$\frac{169}{25}x^{2}-2704=0$$

Divide both sides by \frac{169}{25}.

$$x^{2}-400=0$$

Consider x^{2}-400. Rewrite x^{2}-400 as x^{2}-20^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x-20\right)\left(x+20\right)=0$$

To find equation solutions, solve x-20=0 and x+20=0.

$$x=20 x=-20$$

Steps by Finding Square Root

Combine \frac{144}{25}x^{2} and x^{2} to get \frac{169}{25}x^{2}.

$$\frac{169}{25}x^{2}=52^{2}$$

Calculate 52 to the power of 2 and get 2704.

$$\frac{169}{25}x^{2}=2704$$

Multiply both sides by \frac{25}{169}, the reciprocal of \frac{169}{25}.

$$x^{2}=2704\times \frac{25}{169}$$

Multiply 2704 and \frac{25}{169} to get 400.

$$x^{2}=400$$

Take the square root of both sides of the equation.

$$x=20 x=-20$$

Steps Using the Quadratic Formula

Combine \frac{144}{25}x^{2} and x^{2} to get \frac{169}{25}x^{2}.

$$\frac{169}{25}x^{2}=52^{2}$$

Calculate 52 to the power of 2 and get 2704.

$$\frac{169}{25}x^{2}=2704$$

Subtract 2704 from both sides.

$$\frac{169}{25}x^{2}-2704=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{169}{25} for a, 0 for b, and -2704 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times \frac{169}{25}\left(-2704\right)}}{2\times \frac{169}{25}}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times \frac{169}{25}\left(-2704\right)}}{2\times \frac{169}{25}}$$

Multiply -4 times \frac{169}{25}.

$$x=\frac{0±\sqrt{-\frac{676}{25}\left(-2704\right)}}{2\times \frac{169}{25}}$$

Multiply -\frac{676}{25} times -2704.

$$x=\frac{0±\sqrt{\frac{1827904}{25}}}{2\times \frac{169}{25}}$$

Take the square root of \frac{1827904}{25}.

$$x=\frac{0±\frac{1352}{5}}{2\times \frac{169}{25}}$$

Multiply 2 times \frac{169}{25}.

$$x=\frac{0±\frac{1352}{5}}{\frac{338}{25}}$$

Now solve the equation x=\frac{0±\frac{1352}{5}}{\frac{338}{25}} when ± is plus.

$$x=20$$

Now solve the equation x=\frac{0±\frac{1352}{5}}{\frac{338}{25}} when ± is minus.

$$x=-20$$

The equation is now solved.

$$x=20 x=-20$$