Solve 80b-b^2=1500

$$80 b - b ^ { 2 } = 1500$$

Solve for b

$$b=30$$

$$b=50$$

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.

$$-b^{2}+80b=1500$$

Subtract 1500 from both sides of the equation.

$$-b^{2}+80b-1500=1500-1500$$

Subtracting 1500 from itself leaves 0.

$$-b^{2}+80b-1500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 80 for b, and -1500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$b=\frac{-80±\sqrt{80^{2}-4\left(-1\right)\left(-1500\right)}}{2\left(-1\right)}$$

Square 80.

$$b=\frac{-80±\sqrt{6400-4\left(-1\right)\left(-1500\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$b=\frac{-80±\sqrt{6400+4\left(-1500\right)}}{2\left(-1\right)}$$

Multiply 4 times -1500.

$$b=\frac{-80±\sqrt{6400-6000}}{2\left(-1\right)}$$

Add 6400 to -6000.

$$b=\frac{-80±\sqrt{400}}{2\left(-1\right)}$$

Take the square root of 400.

$$b=\frac{-80±20}{2\left(-1\right)}$$

Multiply 2 times -1.

$$b=\frac{-80±20}{-2}$$

Now solve the equation b=\frac{-80±20}{-2} when ± is plus. Add -80 to 20.

$$b=-\frac{60}{-2}$$

Divide -60 by -2.

$$b=30$$

Now solve the equation b=\frac{-80±20}{-2} when ± is minus. Subtract 20 from -80.

$$b=-\frac{100}{-2}$$

Divide -100 by -2.

$$b=50$$

The equation is now solved.

$$b=30 b=50$$

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.

$$-b^{2}+80b=1500$$

Divide both sides by -1.

$$\frac{-b^{2}+80b}{-1}=\frac{1500}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$b^{2}+\frac{80}{-1}b=\frac{1500}{-1}$$

Divide 80 by -1.

$$b^{2}-80b=\frac{1500}{-1}$$

Divide 1500 by -1.

$$b^{2}-80b=-1500$$

Divide -80, the coefficient of the x term, by 2 to get -40. Then add the square of -40 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$b^{2}-80b+\left(-40\right)^{2}=-1500+\left(-40\right)^{2}$$

Square -40.

$$b^{2}-80b+1600=-1500+1600$$

Add -1500 to 1600.

$$b^{2}-80b+1600=100$$

Factor b^{2}-80b+1600. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(b-40\right)^{2}=100$$

Take the square root of both sides of the equation.

$$\sqrt{\left(b-40\right)^{2}}=\sqrt{100}$$

Simplify.

$$b-40=10 b-40=-10$$

Add 40 to both sides of the equation.

$$b=50 b=30$$