Solve (5/2y+7/2)^2+y^2+3(5/2y+1/2)+4y-1=0

$$( \frac { 5 } { 2 } y + \frac { 7 } { 2 } ) ^ { 2 } + y ^ { 2 } + 3 ( \frac { 5 } { 2 } y + \frac { 1 } { 2 } ) + 4 y - 1 = 0$$

Solve for y

$$y=\frac{\sqrt{1885}}{29}-2\approx -0.502876321$$

$$y=-\frac{\sqrt{1885}}{29}-2\approx -3.497123679$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\frac{5}{2}y+\frac{7}{2}\right)^{2}.

$$\frac{25}{4}y^{2}+\frac{35}{2}y+\frac{49}{4}+y^{2}+3\left(\frac{5}{2}y+\frac{1}{2}\right)+4y-1=0$$

Combine \frac{25}{4}y^{2} and y^{2} to get \frac{29}{4}y^{2}.

$$\frac{29}{4}y^{2}+\frac{35}{2}y+\frac{49}{4}+3\left(\frac{5}{2}y+\frac{1}{2}\right)+4y-1=0$$

Use the distributive property to multiply 3 by \frac{5}{2}y+\frac{1}{2}.

$$\frac{29}{4}y^{2}+\frac{35}{2}y+\frac{49}{4}+\frac{15}{2}y+\frac{3}{2}+4y-1=0$$

Combine \frac{35}{2}y and \frac{15}{2}y to get 25y.

$$\frac{29}{4}y^{2}+25y+\frac{49}{4}+\frac{3}{2}+4y-1=0$$

Add \frac{49}{4} and \frac{3}{2} to get \frac{55}{4}.

$$\frac{29}{4}y^{2}+25y+\frac{55}{4}+4y-1=0$$

Combine 25y and 4y to get 29y.

$$\frac{29}{4}y^{2}+29y+\frac{55}{4}-1=0$$

Subtract 1 from \frac{55}{4} to get \frac{51}{4}.

$$\frac{29}{4}y^{2}+29y+\frac{51}{4}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{29}{4} for a, 29 for b, and \frac{51}{4} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-29±\sqrt{29^{2}-4\times \frac{29}{4}\times \frac{51}{4}}}{2\times \frac{29}{4}}$$

Square 29.

$$y=\frac{-29±\sqrt{841-4\times \frac{29}{4}\times \frac{51}{4}}}{2\times \frac{29}{4}}$$

Multiply -4 times \frac{29}{4}.

$$y=\frac{-29±\sqrt{841-29\times \frac{51}{4}}}{2\times \frac{29}{4}}$$

Multiply -29 times \frac{51}{4}.

$$y=\frac{-29±\sqrt{841-\frac{1479}{4}}}{2\times \frac{29}{4}}$$

Add 841 to -\frac{1479}{4}.

$$y=\frac{-29±\sqrt{\frac{1885}{4}}}{2\times \frac{29}{4}}$$

Take the square root of \frac{1885}{4}.

$$y=\frac{-29±\frac{\sqrt{1885}}{2}}{2\times \frac{29}{4}}$$

Multiply 2 times \frac{29}{4}.

$$y=\frac{-29±\frac{\sqrt{1885}}{2}}{\frac{29}{2}}$$

Now solve the equation y=\frac{-29±\frac{\sqrt{1885}}{2}}{\frac{29}{2}} when ± is plus. Add -29 to \frac{\sqrt{1885}}{2}.

$$y=\frac{\frac{\sqrt{1885}}{2}-29}{\frac{29}{2}}$$

Divide -29+\frac{\sqrt{1885}}{2} by \frac{29}{2} by multiplying -29+\frac{\sqrt{1885}}{2} by the reciprocal of \frac{29}{2}.

$$y=\frac{\sqrt{1885}}{29}-2$$

Now solve the equation y=\frac{-29±\frac{\sqrt{1885}}{2}}{\frac{29}{2}} when ± is minus. Subtract \frac{\sqrt{1885}}{2} from -29.

$$y=\frac{-\frac{\sqrt{1885}}{2}-29}{\frac{29}{2}}$$

Divide -29-\frac{\sqrt{1885}}{2} by \frac{29}{2} by multiplying -29-\frac{\sqrt{1885}}{2} by the reciprocal of \frac{29}{2}.

$$y=-\frac{\sqrt{1885}}{29}-2$$

The equation is now solved.

$$y=\frac{\sqrt{1885}}{29}-2 y=-\frac{\sqrt{1885}}{29}-2$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\frac{5}{2}y+\frac{7}{2}\right)^{2}.

$$\frac{25}{4}y^{2}+\frac{35}{2}y+\frac{49}{4}+y^{2}+3\left(\frac{5}{2}y+\frac{1}{2}\right)+4y-1=0$$

Combine \frac{25}{4}y^{2} and y^{2} to get \frac{29}{4}y^{2}.

$$\frac{29}{4}y^{2}+\frac{35}{2}y+\frac{49}{4}+3\left(\frac{5}{2}y+\frac{1}{2}\right)+4y-1=0$$

Use the distributive property to multiply 3 by \frac{5}{2}y+\frac{1}{2}.

$$\frac{29}{4}y^{2}+\frac{35}{2}y+\frac{49}{4}+\frac{15}{2}y+\frac{3}{2}+4y-1=0$$

Combine \frac{35}{2}y and \frac{15}{2}y to get 25y.

$$\frac{29}{4}y^{2}+25y+\frac{49}{4}+\frac{3}{2}+4y-1=0$$

Add \frac{49}{4} and \frac{3}{2} to get \frac{55}{4}.

$$\frac{29}{4}y^{2}+25y+\frac{55}{4}+4y-1=0$$

Combine 25y and 4y to get 29y.

$$\frac{29}{4}y^{2}+29y+\frac{55}{4}-1=0$$

Subtract 1 from \frac{55}{4} to get \frac{51}{4}.

$$\frac{29}{4}y^{2}+29y+\frac{51}{4}=0$$

Subtract \frac{51}{4} from both sides. Anything subtracted from zero gives its negation.

$$\frac{29}{4}y^{2}+29y=-\frac{51}{4}$$

Divide both sides of the equation by \frac{29}{4}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{29}{4}y^{2}+29y}{\frac{29}{4}}=-\frac{\frac{51}{4}}{\frac{29}{4}}$$

Dividing by \frac{29}{4} undoes the multiplication by \frac{29}{4}.

$$y^{2}+\frac{29}{\frac{29}{4}}y=-\frac{\frac{51}{4}}{\frac{29}{4}}$$

Divide 29 by \frac{29}{4} by multiplying 29 by the reciprocal of \frac{29}{4}.

$$y^{2}+4y=-\frac{\frac{51}{4}}{\frac{29}{4}}$$

Divide -\frac{51}{4} by \frac{29}{4} by multiplying -\frac{51}{4} by the reciprocal of \frac{29}{4}.

$$y^{2}+4y=-\frac{51}{29}$$

Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+4y+2^{2}=-\frac{51}{29}+2^{2}$$

Square 2.

$$y^{2}+4y+4=-\frac{51}{29}+4$$

Add -\frac{51}{29} to 4.

$$y^{2}+4y+4=\frac{65}{29}$$

Factor y^{2}+4y+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y+2\right)^{2}=\frac{65}{29}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+2\right)^{2}}=\sqrt{\frac{65}{29}}$$

Simplify.

$$y+2=\frac{\sqrt{1885}}{29} y+2=-\frac{\sqrt{1885}}{29}$$

Subtract 2 from both sides of the equation.

$$y=\frac{\sqrt{1885}}{29}-2 y=-\frac{\sqrt{1885}}{29}-2$$