Solve (5x)^2-(3x)^2=36

$$( 5 x ) ^ { 2 } - ( 3 x ) ^ { 2 } = 36$$

Solve for x

$$x = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

Steps Using Difference of Squares

Expand \left(5x\right)^{2}.

$$5^{2}x^{2}-\left(3x\right)^{2}=36$$

Calculate 5 to the power of 2 and get 25.

$$25x^{2}-\left(3x\right)^{2}=36$$

Expand \left(3x\right)^{2}.

$$25x^{2}-3^{2}x^{2}=36$$

Calculate 3 to the power of 2 and get 9.

$$25x^{2}-9x^{2}=36$$

Combine 25x^{2} and -9x^{2} to get 16x^{2}.

$$16x^{2}=36$$

Subtract 36 from both sides.

$$16x^{2}-36=0$$

Divide both sides by 4.

$$4x^{2}-9=0$$

Consider 4x^{2}-9. Rewrite 4x^{2}-9 as \left(2x\right)^{2}-3^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(2x-3\right)\left(2x+3\right)=0$$

To find equation solutions, solve 2x-3=0 and 2x+3=0.

$$x=\frac{3}{2} x=-\frac{3}{2}$$

Steps by Finding Square Root

Expand \left(5x\right)^{2}.

$$5^{2}x^{2}-\left(3x\right)^{2}=36$$

Calculate 5 to the power of 2 and get 25.

$$25x^{2}-\left(3x\right)^{2}=36$$

Expand \left(3x\right)^{2}.

$$25x^{2}-3^{2}x^{2}=36$$

Calculate 3 to the power of 2 and get 9.

$$25x^{2}-9x^{2}=36$$

Combine 25x^{2} and -9x^{2} to get 16x^{2}.

$$16x^{2}=36$$

Divide both sides by 16.

$$x^{2}=\frac{36}{16}$$

Reduce the fraction \frac{36}{16} to lowest terms by extracting and canceling out 4.

$$x^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$x=\frac{3}{2} x=-\frac{3}{2}$$

Steps Using the Quadratic Formula

Expand \left(5x\right)^{2}.

$$5^{2}x^{2}-\left(3x\right)^{2}=36$$

Calculate 5 to the power of 2 and get 25.

$$25x^{2}-\left(3x\right)^{2}=36$$

Expand \left(3x\right)^{2}.

$$25x^{2}-3^{2}x^{2}=36$$

Calculate 3 to the power of 2 and get 9.

$$25x^{2}-9x^{2}=36$$

Combine 25x^{2} and -9x^{2} to get 16x^{2}.

$$16x^{2}=36$$

Subtract 36 from both sides.

$$16x^{2}-36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, 0 for b, and -36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 16\left(-36\right)}}{2\times 16}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 16\left(-36\right)}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{0±\sqrt{-64\left(-36\right)}}{2\times 16}$$

Multiply -64 times -36.

$$x=\frac{0±\sqrt{2304}}{2\times 16}$$

Take the square root of 2304.

$$x=\frac{0±48}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{0±48}{32}$$

Now solve the equation x=\frac{0±48}{32} when ± is plus. Reduce the fraction \frac{48}{32} to lowest terms by extracting and canceling out 16.

$$x=\frac{3}{2}$$

Now solve the equation x=\frac{0±48}{32} when ± is minus. Reduce the fraction \frac{-48}{32} to lowest terms by extracting and canceling out 16.

$$x=-\frac{3}{2}$$

The equation is now solved.

$$x=\frac{3}{2} x=-\frac{3}{2}$$