Solve 2/1+t^2=4/3

$$\frac { 2 } { 1 + t ^ { 2 } } = \frac { 4 } { 3 }$$

Solve for t

$$t=\frac{\sqrt{2}}{2}\approx 0.707106781$$

$$t=-\frac{\sqrt{2}}{2}\approx -0.707106781$$

Steps by Finding Square Root

Multiply both sides of the equation by 3\left(t^{2}+1\right), the least common multiple of 1+t^{2},3.

$$3\times 2=4\left(t^{2}+1\right)$$

Multiply 3 and 2 to get 6.

$$6=4\left(t^{2}+1\right)$$

Use the distributive property to multiply 4 by t^{2}+1.

$$6=4t^{2}+4$$

Swap sides so that all variable terms are on the left hand side.

$$4t^{2}+4=6$$

Subtract 4 from both sides.

$$4t^{2}=6-4$$

Subtract 4 from 6 to get 2.

$$4t^{2}=2$$

Divide both sides by 4.

$$t^{2}=\frac{2}{4}$$

Reduce the fraction \frac{2}{4} to lowest terms by extracting and canceling out 2.

$$t^{2}=\frac{1}{2}$$

Take the square root of both sides of the equation.

$$t=\frac{\sqrt{2}}{2} t=-\frac{\sqrt{2}}{2}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 3\left(t^{2}+1\right), the least common multiple of 1+t^{2},3.

$$3\times 2=4\left(t^{2}+1\right)$$

Multiply 3 and 2 to get 6.

$$6=4\left(t^{2}+1\right)$$

Use the distributive property to multiply 4 by t^{2}+1.

$$6=4t^{2}+4$$

Swap sides so that all variable terms are on the left hand side.

$$4t^{2}+4=6$$

Subtract 6 from both sides.

$$4t^{2}+4-6=0$$

Subtract 6 from 4 to get -2.

$$4t^{2}-2=0$$

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

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

Square 0.

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

Multiply -4 times 4.

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

Multiply -16 times -2.

$$t=\frac{0±\sqrt{32}}{2\times 4}$$

Take the square root of 32.

$$t=\frac{0±4\sqrt{2}}{2\times 4}$$

Multiply 2 times 4.

$$t=\frac{0±4\sqrt{2}}{8}$$

Now solve the equation t=\frac{0±4\sqrt{2}}{8} when ± is plus.

$$t=\frac{\sqrt{2}}{2}$$

Now solve the equation t=\frac{0±4\sqrt{2}}{8} when ± is minus.

$$t=-\frac{\sqrt{2}}{2}$$

The equation is now solved.

$$t=\frac{\sqrt{2}}{2} t=-\frac{\sqrt{2}}{2}$$