Solve (12/5)^2+y^2=16

$$( \frac { 12 } { 5 } ) ^ { 2 } + y ^ { 2 } = 16$$

Solve for y

$$y = \frac{16}{5} = 3\frac{1}{5} = 3.2$$

$$y = -\frac{16}{5} = -3\frac{1}{5} = -3.2$$

Steps Using Difference of Squares

Calculate \frac{12}{5} to the power of 2 and get \frac{144}{25}.

$$\frac{144}{25}+y^{2}=16$$

Subtract 16 from both sides.

$$\frac{144}{25}+y^{2}-16=0$$

Subtract 16 from \frac{144}{25} to get -\frac{256}{25}.

$$-\frac{256}{25}+y^{2}=0$$

Multiply both sides by 25.

$$-256+25y^{2}=0$$

Consider -256+25y^{2}. Rewrite -256+25y^{2} as \left(5y\right)^{2}-16^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(5y-16\right)\left(5y+16\right)=0$$

To find equation solutions, solve 5y-16=0 and 5y+16=0.

$$y=\frac{16}{5} y=-\frac{16}{5}$$

Steps by Finding Square Root

Calculate \frac{12}{5} to the power of 2 and get \frac{144}{25}.

$$\frac{144}{25}+y^{2}=16$$

Subtract \frac{144}{25} from both sides.

$$y^{2}=16-\frac{144}{25}$$

Subtract \frac{144}{25} from 16 to get \frac{256}{25}.

$$y^{2}=\frac{256}{25}$$

Take the square root of both sides of the equation.

$$y=\frac{16}{5} y=-\frac{16}{5}$$

Steps Using the Quadratic Formula

Calculate \frac{12}{5} to the power of 2 and get \frac{144}{25}.

$$\frac{144}{25}+y^{2}=16$$

Subtract 16 from both sides.

$$\frac{144}{25}+y^{2}-16=0$$

Subtract 16 from \frac{144}{25} to get -\frac{256}{25}.

$$-\frac{256}{25}+y^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$y^{2}-\frac{256}{25}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and -\frac{256}{25} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{0±\sqrt{0^{2}-4\left(-\frac{256}{25}\right)}}{2}$$

Square 0.

$$y=\frac{0±\sqrt{-4\left(-\frac{256}{25}\right)}}{2}$$

Multiply -4 times -\frac{256}{25}.

$$y=\frac{0±\sqrt{\frac{1024}{25}}}{2}$$

Take the square root of \frac{1024}{25}.

$$y=\frac{0±\frac{32}{5}}{2}$$

Now solve the equation y=\frac{0±\frac{32}{5}}{2} when ± is plus.

$$y=\frac{16}{5}$$

Now solve the equation y=\frac{0±\frac{32}{5}}{2} when ± is minus.

$$y=-\frac{16}{5}$$

The equation is now solved.

$$y=\frac{16}{5} y=-\frac{16}{5}$$