Solve 4y^2-56y=108

$$4 y ^ { 2 } - 56 y = 108$$

Solve for y

$$y=2\sqrt{19}+7\approx 15.717797887$$

$$y=7-2\sqrt{19}\approx -1.717797887$$

Steps Using the Quadratic Formula

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.

$$4y^{2}-56y=108$$

Subtract 108 from both sides of the equation.

$$4y^{2}-56y-108=108-108$$

Subtracting 108 from itself leaves 0.

$$4y^{2}-56y-108=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -56 for b, and -108 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-56\right)±\sqrt{\left(-56\right)^{2}-4\times 4\left(-108\right)}}{2\times 4}$$

Square -56.

$$y=\frac{-\left(-56\right)±\sqrt{3136-4\times 4\left(-108\right)}}{2\times 4}$$

Multiply -4 times 4.

$$y=\frac{-\left(-56\right)±\sqrt{3136-16\left(-108\right)}}{2\times 4}$$

Multiply -16 times -108.

$$y=\frac{-\left(-56\right)±\sqrt{3136+1728}}{2\times 4}$$

Add 3136 to 1728.

$$y=\frac{-\left(-56\right)±\sqrt{4864}}{2\times 4}$$

Take the square root of 4864.

$$y=\frac{-\left(-56\right)±16\sqrt{19}}{2\times 4}$$

The opposite of -56 is 56.

$$y=\frac{56±16\sqrt{19}}{2\times 4}$$

Multiply 2 times 4.

$$y=\frac{56±16\sqrt{19}}{8}$$

Now solve the equation y=\frac{56±16\sqrt{19}}{8} when ± is plus. Add 56 to 16\sqrt{19}.

$$y=\frac{16\sqrt{19}+56}{8}$$

Divide 56+16\sqrt{19} by 8.

$$y=2\sqrt{19}+7$$

Now solve the equation y=\frac{56±16\sqrt{19}}{8} when ± is minus. Subtract 16\sqrt{19} from 56.

$$y=\frac{56-16\sqrt{19}}{8}$$

Divide 56-16\sqrt{19} by 8.

$$y=7-2\sqrt{19}$$

The equation is now solved.

$$y=2\sqrt{19}+7 y=7-2\sqrt{19}$$

Steps for Completing the Square

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.

$$4y^{2}-56y=108$$

Divide both sides by 4.

$$\frac{4y^{2}-56y}{4}=\frac{108}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$y^{2}+\left(-\frac{56}{4}\right)y=\frac{108}{4}$$

Divide -56 by 4.

$$y^{2}-14y=\frac{108}{4}$$

Divide 108 by 4.

$$y^{2}-14y=27$$

Divide -14, the coefficient of the x term, by 2 to get -7. Then add the square of -7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-14y+\left(-7\right)^{2}=27+\left(-7\right)^{2}$$

Square -7.

$$y^{2}-14y+49=27+49$$

Add 27 to 49.

$$y^{2}-14y+49=76$$

Factor y^{2}-14y+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-7\right)^{2}=76$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-7\right)^{2}}=\sqrt{76}$$

Simplify.

$$y-7=2\sqrt{19} y-7=-2\sqrt{19}$$

Add 7 to both sides of the equation.

$$y=2\sqrt{19}+7 y=7-2\sqrt{19}$$