Solve 15x(115-x)=750

$$15 x ( 115 - x ) = 750$$

Solve for x

$$x = \frac{5 \sqrt{521} + 115}{2} \approx 114.563561053$$

$$x=\frac{115-5\sqrt{521}}{2}\approx 0.436438947$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 15x by 115-x.

$$1725x-15x^{2}=750$$

Subtract 750 from both sides.

$$1725x-15x^{2}-750=0$$

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.

$$-15x^{2}+1725x-750=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -15 for a, 1725 for b, and -750 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1725±\sqrt{1725^{2}-4\left(-15\right)\left(-750\right)}}{2\left(-15\right)}$$

Square 1725.

$$x=\frac{-1725±\sqrt{2975625-4\left(-15\right)\left(-750\right)}}{2\left(-15\right)}$$

Multiply -4 times -15.

$$x=\frac{-1725±\sqrt{2975625+60\left(-750\right)}}{2\left(-15\right)}$$

Multiply 60 times -750.

$$x=\frac{-1725±\sqrt{2975625-45000}}{2\left(-15\right)}$$

Add 2975625 to -45000.

$$x=\frac{-1725±\sqrt{2930625}}{2\left(-15\right)}$$

Take the square root of 2930625.

$$x=\frac{-1725±75\sqrt{521}}{2\left(-15\right)}$$

Multiply 2 times -15.

$$x=\frac{-1725±75\sqrt{521}}{-30}$$

Now solve the equation x=\frac{-1725±75\sqrt{521}}{-30} when ± is plus. Add -1725 to 75\sqrt{521}.

$$x=\frac{75\sqrt{521}-1725}{-30}$$

Divide -1725+75\sqrt{521} by -30.

$$x=\frac{115-5\sqrt{521}}{2}$$

Now solve the equation x=\frac{-1725±75\sqrt{521}}{-30} when ± is minus. Subtract 75\sqrt{521} from -1725.

$$x=\frac{-75\sqrt{521}-1725}{-30}$$

Divide -1725-75\sqrt{521} by -30.

$$x=\frac{5\sqrt{521}+115}{2}$$

The equation is now solved.

$$x=\frac{115-5\sqrt{521}}{2} x=\frac{5\sqrt{521}+115}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 15x by 115-x.

$$1725x-15x^{2}=750$$

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.

$$-15x^{2}+1725x=750$$

Divide both sides by -15.

$$\frac{-15x^{2}+1725x}{-15}=\frac{750}{-15}$$

Dividing by -15 undoes the multiplication by -15.

$$x^{2}+\frac{1725}{-15}x=\frac{750}{-15}$$

Divide 1725 by -15.

$$x^{2}-115x=\frac{750}{-15}$$

Divide 750 by -15.

$$x^{2}-115x=-50$$

Divide -115, the coefficient of the x term, by 2 to get -\frac{115}{2}. Then add the square of -\frac{115}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-115x+\left(-\frac{115}{2}\right)^{2}=-50+\left(-\frac{115}{2}\right)^{2}$$

Square -\frac{115}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-115x+\frac{13225}{4}=-50+\frac{13225}{4}$$

Add -50 to \frac{13225}{4}.

$$x^{2}-115x+\frac{13225}{4}=\frac{13025}{4}$$

Factor x^{2}-115x+\frac{13225}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{115}{2}\right)^{2}=\frac{13025}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{115}{2}\right)^{2}}=\sqrt{\frac{13025}{4}}$$

Simplify.

$$x-\frac{115}{2}=\frac{5\sqrt{521}}{2} x-\frac{115}{2}=-\frac{5\sqrt{521}}{2}$$

Add \frac{115}{2} to both sides of the equation.

$$x=\frac{5\sqrt{521}+115}{2} x=\frac{115-5\sqrt{521}}{2}$$