Solve (2/3y+7/3)^2+y^2=52

$$( \frac { 2 } { 3 } y + \frac { 7 } { 3 } ) ^ { 2 } + y ^ { 2 } = 52$$

Solve for y

$$y = \frac{3 \sqrt{627} - 14}{13} \approx 4.701531089$$

$$y=\frac{-3\sqrt{627}-14}{13}\approx -6.855377243$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\frac{2}{3}y+\frac{7}{3}\right)^{2}.

$$\frac{4}{9}y^{2}+\frac{28}{9}y+\frac{49}{9}+y^{2}=52$$

Combine \frac{4}{9}y^{2} and y^{2} to get \frac{13}{9}y^{2}.

$$\frac{13}{9}y^{2}+\frac{28}{9}y+\frac{49}{9}=52$$

Subtract 52 from both sides.

$$\frac{13}{9}y^{2}+\frac{28}{9}y+\frac{49}{9}-52=0$$

Subtract 52 from \frac{49}{9} to get -\frac{419}{9}.

$$\frac{13}{9}y^{2}+\frac{28}{9}y-\frac{419}{9}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{13}{9} for a, \frac{28}{9} for b, and -\frac{419}{9} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\frac{28}{9}±\sqrt{\left(\frac{28}{9}\right)^{2}-4\times \frac{13}{9}\left(-\frac{419}{9}\right)}}{2\times \frac{13}{9}}$$

Square \frac{28}{9} by squaring both the numerator and the denominator of the fraction.

$$y=\frac{-\frac{28}{9}±\sqrt{\frac{784}{81}-4\times \frac{13}{9}\left(-\frac{419}{9}\right)}}{2\times \frac{13}{9}}$$

Multiply -4 times \frac{13}{9}.

$$y=\frac{-\frac{28}{9}±\sqrt{\frac{784}{81}-\frac{52}{9}\left(-\frac{419}{9}\right)}}{2\times \frac{13}{9}}$$

Multiply -\frac{52}{9} times -\frac{419}{9} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\frac{28}{9}±\sqrt{\frac{784+21788}{81}}}{2\times \frac{13}{9}}$$

Add \frac{784}{81} to \frac{21788}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\frac{28}{9}±\sqrt{\frac{836}{3}}}{2\times \frac{13}{9}}$$

Take the square root of \frac{836}{3}.

$$y=\frac{-\frac{28}{9}±\frac{2\sqrt{627}}{3}}{2\times \frac{13}{9}}$$

Multiply 2 times \frac{13}{9}.

$$y=\frac{-\frac{28}{9}±\frac{2\sqrt{627}}{3}}{\frac{26}{9}}$$

Now solve the equation y=\frac{-\frac{28}{9}±\frac{2\sqrt{627}}{3}}{\frac{26}{9}} when ± is plus. Add -\frac{28}{9} to \frac{2\sqrt{627}}{3}.

$$y=\frac{\frac{2\sqrt{627}}{3}-\frac{28}{9}}{\frac{26}{9}}$$

Divide -\frac{28}{9}+\frac{2\sqrt{627}}{3} by \frac{26}{9} by multiplying -\frac{28}{9}+\frac{2\sqrt{627}}{3} by the reciprocal of \frac{26}{9}.

$$y=\frac{3\sqrt{627}-14}{13}$$

Now solve the equation y=\frac{-\frac{28}{9}±\frac{2\sqrt{627}}{3}}{\frac{26}{9}} when ± is minus. Subtract \frac{2\sqrt{627}}{3} from -\frac{28}{9}.

$$y=\frac{-\frac{2\sqrt{627}}{3}-\frac{28}{9}}{\frac{26}{9}}$$

Divide -\frac{28}{9}-\frac{2\sqrt{627}}{3} by \frac{26}{9} by multiplying -\frac{28}{9}-\frac{2\sqrt{627}}{3} by the reciprocal of \frac{26}{9}.

$$y=\frac{-3\sqrt{627}-14}{13}$$

The equation is now solved.

$$y=\frac{3\sqrt{627}-14}{13} y=\frac{-3\sqrt{627}-14}{13}$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\frac{2}{3}y+\frac{7}{3}\right)^{2}.

$$\frac{4}{9}y^{2}+\frac{28}{9}y+\frac{49}{9}+y^{2}=52$$

Combine \frac{4}{9}y^{2} and y^{2} to get \frac{13}{9}y^{2}.

$$\frac{13}{9}y^{2}+\frac{28}{9}y+\frac{49}{9}=52$$

Subtract \frac{49}{9} from both sides.

$$\frac{13}{9}y^{2}+\frac{28}{9}y=52-\frac{49}{9}$$

Subtract \frac{49}{9} from 52 to get \frac{419}{9}.

$$\frac{13}{9}y^{2}+\frac{28}{9}y=\frac{419}{9}$$

Divide both sides of the equation by \frac{13}{9}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{13}{9}y^{2}+\frac{28}{9}y}{\frac{13}{9}}=\frac{\frac{419}{9}}{\frac{13}{9}}$$

Dividing by \frac{13}{9} undoes the multiplication by \frac{13}{9}.

$$y^{2}+\frac{\frac{28}{9}}{\frac{13}{9}}y=\frac{\frac{419}{9}}{\frac{13}{9}}$$

Divide \frac{28}{9} by \frac{13}{9} by multiplying \frac{28}{9} by the reciprocal of \frac{13}{9}.

$$y^{2}+\frac{28}{13}y=\frac{\frac{419}{9}}{\frac{13}{9}}$$

Divide \frac{419}{9} by \frac{13}{9} by multiplying \frac{419}{9} by the reciprocal of \frac{13}{9}.

$$y^{2}+\frac{28}{13}y=\frac{419}{13}$$

Divide \frac{28}{13}, the coefficient of the x term, by 2 to get \frac{14}{13}. Then add the square of \frac{14}{13} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+\frac{28}{13}y+\left(\frac{14}{13}\right)^{2}=\frac{419}{13}+\left(\frac{14}{13}\right)^{2}$$

Square \frac{14}{13} by squaring both the numerator and the denominator of the fraction.

$$y^{2}+\frac{28}{13}y+\frac{196}{169}=\frac{419}{13}+\frac{196}{169}$$

Add \frac{419}{13} to \frac{196}{169} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}+\frac{28}{13}y+\frac{196}{169}=\frac{5643}{169}$$

Factor y^{2}+\frac{28}{13}y+\frac{196}{169}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y+\frac{14}{13}\right)^{2}=\frac{5643}{169}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+\frac{14}{13}\right)^{2}}=\sqrt{\frac{5643}{169}}$$

Simplify.

$$y+\frac{14}{13}=\frac{3\sqrt{627}}{13} y+\frac{14}{13}=-\frac{3\sqrt{627}}{13}$$

Subtract \frac{14}{13} from both sides of the equation.

$$y=\frac{3\sqrt{627}-14}{13} y=\frac{-3\sqrt{627}-14}{13}$$