Solve 5032y-4y^2=40000

$$5032 y - 4 y ^ { 2 } = 40000$$

Solve for y

$$y=8$$

$$y=1250$$

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.

$$-4y^{2}+5032y=40000$$

Subtract 40000 from both sides of the equation.

$$-4y^{2}+5032y-40000=40000-40000$$

Subtracting 40000 from itself leaves 0.

$$-4y^{2}+5032y-40000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 5032 for b, and -40000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-5032±\sqrt{5032^{2}-4\left(-4\right)\left(-40000\right)}}{2\left(-4\right)}$$

Square 5032.

$$y=\frac{-5032±\sqrt{25321024-4\left(-4\right)\left(-40000\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$y=\frac{-5032±\sqrt{25321024+16\left(-40000\right)}}{2\left(-4\right)}$$

Multiply 16 times -40000.

$$y=\frac{-5032±\sqrt{25321024-640000}}{2\left(-4\right)}$$

Add 25321024 to -640000.

$$y=\frac{-5032±\sqrt{24681024}}{2\left(-4\right)}$$

Take the square root of 24681024.

$$y=\frac{-5032±4968}{2\left(-4\right)}$$

Multiply 2 times -4.

$$y=\frac{-5032±4968}{-8}$$

Now solve the equation y=\frac{-5032±4968}{-8} when ± is plus. Add -5032 to 4968.

$$y=-\frac{64}{-8}$$

Divide -64 by -8.

$$y=8$$

Now solve the equation y=\frac{-5032±4968}{-8} when ± is minus. Subtract 4968 from -5032.

$$y=-\frac{10000}{-8}$$

Divide -10000 by -8.

$$y=1250$$

The equation is now solved.

$$y=8 y=1250$$

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.

$$-4y^{2}+5032y=40000$$

Divide both sides by -4.

$$\frac{-4y^{2}+5032y}{-4}=\frac{40000}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$y^{2}+\frac{5032}{-4}y=\frac{40000}{-4}$$

Divide 5032 by -4.

$$y^{2}-1258y=\frac{40000}{-4}$$

Divide 40000 by -4.

$$y^{2}-1258y=-10000$$

Divide -1258, the coefficient of the x term, by 2 to get -629. Then add the square of -629 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-1258y+\left(-629\right)^{2}=-10000+\left(-629\right)^{2}$$

Square -629.

$$y^{2}-1258y+395641=-10000+395641$$

Add -10000 to 395641.

$$y^{2}-1258y+395641=385641$$

Factor y^{2}-1258y+395641. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-629\right)^{2}=385641$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-629\right)^{2}}=\sqrt{385641}$$

Simplify.

$$y-629=621 y-629=-621$$

Add 629 to both sides of the equation.

$$y=1250 y=8$$