Solve (30+x)(1000-3x)-30000-310x=30000

$$(30+x)(1000-3x)-30000-310x=30000$$

Solve for x

$$x=100$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 30+x by 1000-3x and combine like terms.

$$30000+910x-3x^{2}-30000-310x=30000$$

Subtract 30000 from 30000 to get 0.

$$910x-3x^{2}-310x=30000$$

Combine 910x and -310x to get 600x.

$$600x-3x^{2}=30000$$

Subtract 30000 from both sides.

$$600x-3x^{2}-30000=0$$

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.

$$-3x^{2}+600x-30000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 600 for b, and -30000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-600±\sqrt{600^{2}-4\left(-3\right)\left(-30000\right)}}{2\left(-3\right)}$$

Square 600.

$$x=\frac{-600±\sqrt{360000-4\left(-3\right)\left(-30000\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-600±\sqrt{360000+12\left(-30000\right)}}{2\left(-3\right)}$$

Multiply 12 times -30000.

$$x=\frac{-600±\sqrt{360000-360000}}{2\left(-3\right)}$$

Add 360000 to -360000.

$$x=\frac{-600±\sqrt{0}}{2\left(-3\right)}$$

Take the square root of 0.

$$x=-\frac{600}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=-\frac{600}{-6}$$

Divide -600 by -6.

$$x=100$$

Steps for Completing the Square

Use the distributive property to multiply 30+x by 1000-3x and combine like terms.

$$30000+910x-3x^{2}-30000-310x=30000$$

Subtract 30000 from 30000 to get 0.

$$910x-3x^{2}-310x=30000$$

Combine 910x and -310x to get 600x.

$$600x-3x^{2}=30000$$

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.

$$-3x^{2}+600x=30000$$

Divide both sides by -3.

$$\frac{-3x^{2}+600x}{-3}=\frac{30000}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{600}{-3}x=\frac{30000}{-3}$$

Divide 600 by -3.

$$x^{2}-200x=\frac{30000}{-3}$$

Divide 30000 by -3.

$$x^{2}-200x=-10000$$

Divide -200, the coefficient of the x term, by 2 to get -100. Then add the square of -100 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-200x+\left(-100\right)^{2}=-10000+\left(-100\right)^{2}$$

Square -100.

$$x^{2}-200x+10000=-10000+10000$$

Add -10000 to 10000.

$$x^{2}-200x+10000=0$$

Factor x^{2}-200x+10000. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-100\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-100\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-100=0 x-100=0$$

Add 100 to both sides of the equation.

$$x=100 x=100$$

The equation is now solved. Solutions are the same.

$$x=100$$