Solve y^2-8y/25-1673/25=0

$$y ^ { 2 } - \frac { 8 y } { 25 } - \frac { 1673 } { 25 } = 0$$

Solve for y

$$y = \frac{3 \sqrt{4649} + 4}{25} \approx 8.342029088$$

$$y=\frac{4-3\sqrt{4649}}{25}\approx -8.022029088$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 25.

$$25y^{2}-8y-1673=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 25 for a, -8 for b, and -1673 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 25\left(-1673\right)}}{2\times 25}$$

Square -8.

$$y=\frac{-\left(-8\right)±\sqrt{64-4\times 25\left(-1673\right)}}{2\times 25}$$

Multiply -4 times 25.

$$y=\frac{-\left(-8\right)±\sqrt{64-100\left(-1673\right)}}{2\times 25}$$

Multiply -100 times -1673.

$$y=\frac{-\left(-8\right)±\sqrt{64+167300}}{2\times 25}$$

Add 64 to 167300.

$$y=\frac{-\left(-8\right)±\sqrt{167364}}{2\times 25}$$

Take the square root of 167364.

$$y=\frac{-\left(-8\right)±6\sqrt{4649}}{2\times 25}$$

The opposite of -8 is 8.

$$y=\frac{8±6\sqrt{4649}}{2\times 25}$$

Multiply 2 times 25.

$$y=\frac{8±6\sqrt{4649}}{50}$$

Now solve the equation y=\frac{8±6\sqrt{4649}}{50} when ± is plus. Add 8 to 6\sqrt{4649}.

$$y=\frac{6\sqrt{4649}+8}{50}$$

Divide 8+6\sqrt{4649} by 50.

$$y=\frac{3\sqrt{4649}+4}{25}$$

Now solve the equation y=\frac{8±6\sqrt{4649}}{50} when ± is minus. Subtract 6\sqrt{4649} from 8.

$$y=\frac{8-6\sqrt{4649}}{50}$$

Divide 8-6\sqrt{4649} by 50.

$$y=\frac{4-3\sqrt{4649}}{25}$$

The equation is now solved.

$$y=\frac{3\sqrt{4649}+4}{25} y=\frac{4-3\sqrt{4649}}{25}$$

Steps for Completing the Square

Multiply both sides of the equation by 25.

$$25y^{2}-8y-1673=0$$

Add 1673 to both sides. Anything plus zero gives itself.

$$25y^{2}-8y=1673$$

Divide both sides by 25.

$$\frac{25y^{2}-8y}{25}=\frac{1673}{25}$$

Dividing by 25 undoes the multiplication by 25.

$$y^{2}-\frac{8}{25}y=\frac{1673}{25}$$

Divide -\frac{8}{25}, the coefficient of the x term, by 2 to get -\frac{4}{25}. Then add the square of -\frac{4}{25} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-\frac{8}{25}y+\left(-\frac{4}{25}\right)^{2}=\frac{1673}{25}+\left(-\frac{4}{25}\right)^{2}$$

Square -\frac{4}{25} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-\frac{8}{25}y+\frac{16}{625}=\frac{1673}{25}+\frac{16}{625}$$

Add \frac{1673}{25} to \frac{16}{625} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}-\frac{8}{25}y+\frac{16}{625}=\frac{41841}{625}$$

Factor y^{2}-\frac{8}{25}y+\frac{16}{625}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-\frac{4}{25}\right)^{2}=\frac{41841}{625}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{4}{25}\right)^{2}}=\sqrt{\frac{41841}{625}}$$

Simplify.

$$y-\frac{4}{25}=\frac{3\sqrt{4649}}{25} y-\frac{4}{25}=-\frac{3\sqrt{4649}}{25}$$

Add \frac{4}{25} to both sides of the equation.

$$y=\frac{3\sqrt{4649}+4}{25} y=\frac{4-3\sqrt{4649}}{25}$$