Solve 25y=49+4y^2-56y

$$25 y = 49 + 4 y ^ { 2 } - 56 y$$

Solve for y

$$y = \frac{\sqrt{5777} + 81}{8} \approx 19.625822333$$

$$y=\frac{81-\sqrt{5777}}{8}\approx 0.624177667$$

Steps Using the Quadratic Formula

Subtract 49 from both sides.

$$25y-49=4y^{2}-56y$$

Subtract 4y^{2} from both sides.

$$25y-49-4y^{2}=-56y$$

Add 56y to both sides.

$$25y-49-4y^{2}+56y=0$$

Combine 25y and 56y to get 81y.

$$81y-49-4y^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-4y^{2}+81y-49=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 81 for b, and -49 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-81±\sqrt{81^{2}-4\left(-4\right)\left(-49\right)}}{2\left(-4\right)}$$

Square 81.

$$y=\frac{-81±\sqrt{6561-4\left(-4\right)\left(-49\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$y=\frac{-81±\sqrt{6561+16\left(-49\right)}}{2\left(-4\right)}$$

Multiply 16 times -49.

$$y=\frac{-81±\sqrt{6561-784}}{2\left(-4\right)}$$

Add 6561 to -784.

$$y=\frac{-81±\sqrt{5777}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$y=\frac{-81±\sqrt{5777}}{-8}$$

Now solve the equation y=\frac{-81±\sqrt{5777}}{-8} when ± is plus. Add -81 to \sqrt{5777}.

$$y=\frac{\sqrt{5777}-81}{-8}$$

Divide -81+\sqrt{5777} by -8.

$$y=\frac{81-\sqrt{5777}}{8}$$

Now solve the equation y=\frac{-81±\sqrt{5777}}{-8} when ± is minus. Subtract \sqrt{5777} from -81.

$$y=\frac{-\sqrt{5777}-81}{-8}$$

Divide -81-\sqrt{5777} by -8.

$$y=\frac{\sqrt{5777}+81}{8}$$

The equation is now solved.

$$y=\frac{81-\sqrt{5777}}{8} y=\frac{\sqrt{5777}+81}{8}$$

Steps for Completing the Square

Subtract 4y^{2} from both sides.

$$25y-4y^{2}=49-56y$$

Add 56y to both sides.

$$25y-4y^{2}+56y=49$$

Combine 25y and 56y to get 81y.

$$81y-4y^{2}=49$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-4y^{2}+81y=49$$

Divide both sides by -4.

$$\frac{-4y^{2}+81y}{-4}=\frac{49}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$y^{2}+\frac{81}{-4}y=\frac{49}{-4}$$

Divide 81 by -4.

$$y^{2}-\frac{81}{4}y=\frac{49}{-4}$$

Divide 49 by -4.

$$y^{2}-\frac{81}{4}y=-\frac{49}{4}$$

Divide -\frac{81}{4}, the coefficient of the x term, by 2 to get -\frac{81}{8}. Then add the square of -\frac{81}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-\frac{81}{4}y+\left(-\frac{81}{8}\right)^{2}=-\frac{49}{4}+\left(-\frac{81}{8}\right)^{2}$$

Square -\frac{81}{8} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-\frac{81}{4}y+\frac{6561}{64}=-\frac{49}{4}+\frac{6561}{64}$$

Add -\frac{49}{4} to \frac{6561}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}-\frac{81}{4}y+\frac{6561}{64}=\frac{5777}{64}$$

Factor y^{2}-\frac{81}{4}y+\frac{6561}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-\frac{81}{8}\right)^{2}=\frac{5777}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{81}{8}\right)^{2}}=\sqrt{\frac{5777}{64}}$$

Simplify.

$$y-\frac{81}{8}=\frac{\sqrt{5777}}{8} y-\frac{81}{8}=-\frac{\sqrt{5777}}{8}$$

Add \frac{81}{8} to both sides of the equation.

$$y=\frac{\sqrt{5777}+81}{8} y=\frac{81-\sqrt{5777}}{8}$$