Solve 3z^2+7/4+z^2=2

$$\frac { 3 z ^ { 2 } + 7 } { 4 + z ^ { 2 } } = 2$$

Solve for z

$$z=1$$

$$z=-1$$

Steps Using Difference of Squares

Multiply both sides of the equation by z^{2}+4.

$$3z^{2}+7=2\left(z^{2}+4\right)$$

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

$$3z^{2}+7=2z^{2}+8$$

Subtract 2z^{2} from both sides.

$$3z^{2}+7-2z^{2}=8$$

Combine 3z^{2} and -2z^{2} to get z^{2}.

$$z^{2}+7=8$$

Subtract 8 from both sides.

$$z^{2}+7-8=0$$

Subtract 8 from 7 to get -1.

$$z^{2}-1=0$$

Consider z^{2}-1. Rewrite z^{2}-1 as z^{2}-1^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(z-1\right)\left(z+1\right)=0$$

To find equation solutions, solve z-1=0 and z+1=0.

$$z=1 z=-1$$

Steps by Finding Square Root

Multiply both sides of the equation by z^{2}+4.

$$3z^{2}+7=2\left(z^{2}+4\right)$$

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

$$3z^{2}+7=2z^{2}+8$$

Subtract 2z^{2} from both sides.

$$3z^{2}+7-2z^{2}=8$$

Combine 3z^{2} and -2z^{2} to get z^{2}.

$$z^{2}+7=8$$

Subtract 7 from both sides.

$$z^{2}=8-7$$

Subtract 7 from 8 to get 1.

$$z^{2}=1$$

Take the square root of both sides of the equation.

$$z=1 z=-1$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by z^{2}+4.

$$3z^{2}+7=2\left(z^{2}+4\right)$$

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

$$3z^{2}+7=2z^{2}+8$$

Subtract 2z^{2} from both sides.

$$3z^{2}+7-2z^{2}=8$$

Combine 3z^{2} and -2z^{2} to get z^{2}.

$$z^{2}+7=8$$

Subtract 8 from both sides.

$$z^{2}+7-8=0$$

Subtract 8 from 7 to get -1.

$$z^{2}-1=0$$

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

$$z=\frac{0±\sqrt{0^{2}-4\left(-1\right)}}{2}$$

Square 0.

$$z=\frac{0±\sqrt{-4\left(-1\right)}}{2}$$

Multiply -4 times -1.

$$z=\frac{0±\sqrt{4}}{2}$$

Take the square root of 4.

$$z=\frac{0±2}{2}$$

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

$$z=1$$

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

$$z=-1$$

The equation is now solved.

$$z=1 z=-1$$