Solve (24-y)y=128

$$( 24 - y ) y = 128$$

Solve for y

$$y=8$$

$$y=16$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 24-y by y.

$$24y-y^{2}=128$$

Subtract 128 from both sides.

$$24y-y^{2}-128=0$$

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.

$$-y^{2}+24y-128=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 24 for b, and -128 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-24±\sqrt{24^{2}-4\left(-1\right)\left(-128\right)}}{2\left(-1\right)}$$

Square 24.

$$y=\frac{-24±\sqrt{576-4\left(-1\right)\left(-128\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$y=\frac{-24±\sqrt{576+4\left(-128\right)}}{2\left(-1\right)}$$

Multiply 4 times -128.

$$y=\frac{-24±\sqrt{576-512}}{2\left(-1\right)}$$

Add 576 to -512.

$$y=\frac{-24±\sqrt{64}}{2\left(-1\right)}$$

Take the square root of 64.

$$y=\frac{-24±8}{2\left(-1\right)}$$

Multiply 2 times -1.

$$y=\frac{-24±8}{-2}$$

Now solve the equation y=\frac{-24±8}{-2} when ± is plus. Add -24 to 8.

$$y=-\frac{16}{-2}$$

Divide -16 by -2.

$$y=8$$

Now solve the equation y=\frac{-24±8}{-2} when ± is minus. Subtract 8 from -24.

$$y=-\frac{32}{-2}$$

Divide -32 by -2.

$$y=16$$

The equation is now solved.

$$y=8 y=16$$

Steps for Completing the Square

Use the distributive property to multiply 24-y by y.

$$24y-y^{2}=128$$

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.

$$-y^{2}+24y=128$$

Divide both sides by -1.

$$\frac{-y^{2}+24y}{-1}=\frac{128}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$y^{2}+\frac{24}{-1}y=\frac{128}{-1}$$

Divide 24 by -1.

$$y^{2}-24y=\frac{128}{-1}$$

Divide 128 by -1.

$$y^{2}-24y=-128$$

Divide -24, the coefficient of the x term, by 2 to get -12. Then add the square of -12 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-24y+\left(-12\right)^{2}=-128+\left(-12\right)^{2}$$

Square -12.

$$y^{2}-24y+144=-128+144$$

Add -128 to 144.

$$y^{2}-24y+144=16$$

Factor y^{2}-24y+144. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-12\right)^{2}=16$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-12\right)^{2}}=\sqrt{16}$$

Simplify.

$$y-12=4 y-12=-4$$

Add 12 to both sides of the equation.

$$y=16 y=8$$