Solve (10+x)(300-10x)=3750

$$(10+x)(300-10x)=3750$$

Solve for x

$$x=5$$

$$x=15$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 10+x by 300-10x and combine like terms.

$$3000+200x-10x^{2}=3750$$

Subtract 3750 from both sides.

$$3000+200x-10x^{2}-3750=0$$

Subtract 3750 from 3000 to get -750.

$$-750+200x-10x^{2}=0$$

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.

$$-10x^{2}+200x-750=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 200 for b, and -750 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-200±\sqrt{200^{2}-4\left(-10\right)\left(-750\right)}}{2\left(-10\right)}$$

Square 200.

$$x=\frac{-200±\sqrt{40000-4\left(-10\right)\left(-750\right)}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-200±\sqrt{40000+40\left(-750\right)}}{2\left(-10\right)}$$

Multiply 40 times -750.

$$x=\frac{-200±\sqrt{40000-30000}}{2\left(-10\right)}$$

Add 40000 to -30000.

$$x=\frac{-200±\sqrt{10000}}{2\left(-10\right)}$$

Take the square root of 10000.

$$x=\frac{-200±100}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{-200±100}{-20}$$

Now solve the equation x=\frac{-200±100}{-20} when ± is plus. Add -200 to 100.

$$x=-\frac{100}{-20}$$

Divide -100 by -20.

$$x=5$$

Now solve the equation x=\frac{-200±100}{-20} when ± is minus. Subtract 100 from -200.

$$x=-\frac{300}{-20}$$

Divide -300 by -20.

$$x=15$$

The equation is now solved.

$$x=5 x=15$$

Steps for Completing the Square

Use the distributive property to multiply 10+x by 300-10x and combine like terms.

$$3000+200x-10x^{2}=3750$$

Subtract 3000 from both sides.

$$200x-10x^{2}=3750-3000$$

Subtract 3000 from 3750 to get 750.

$$200x-10x^{2}=750$$

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.

$$-10x^{2}+200x=750$$

Divide both sides by -10.

$$\frac{-10x^{2}+200x}{-10}=\frac{750}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\frac{200}{-10}x=\frac{750}{-10}$$

Divide 200 by -10.

$$x^{2}-20x=\frac{750}{-10}$$

Divide 750 by -10.

$$x^{2}-20x=-75$$

Divide -20, the coefficient of the x term, by 2 to get -10. Then add the square of -10 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-20x+\left(-10\right)^{2}=-75+\left(-10\right)^{2}$$

Square -10.

$$x^{2}-20x+100=-75+100$$

Add -75 to 100.

$$x^{2}-20x+100=25$$

Factor x^{2}-20x+100. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-10\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-10\right)^{2}}=\sqrt{25}$$

Simplify.

$$x-10=5 x-10=-5$$

Add 10 to both sides of the equation.

$$x=15 x=5$$