Solve (100-x)*(30+x)=4200

$$(100-x) \times (30+x)=4200$$

Solve for x

$$x=30$$

$$x=40$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 100-x by 30+x and combine like terms.

$$3000+70x-x^{2}=4200$$

Subtract 4200 from both sides.

$$3000+70x-x^{2}-4200=0$$

Subtract 4200 from 3000 to get -1200.

$$-1200+70x-x^{2}=0$$

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.

$$-x^{2}+70x-1200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 70 for b, and -1200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-70±\sqrt{70^{2}-4\left(-1\right)\left(-1200\right)}}{2\left(-1\right)}$$

Square 70.

$$x=\frac{-70±\sqrt{4900-4\left(-1\right)\left(-1200\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-70±\sqrt{4900+4\left(-1200\right)}}{2\left(-1\right)}$$

Multiply 4 times -1200.

$$x=\frac{-70±\sqrt{4900-4800}}{2\left(-1\right)}$$

Add 4900 to -4800.

$$x=\frac{-70±\sqrt{100}}{2\left(-1\right)}$$

Take the square root of 100.

$$x=\frac{-70±10}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-70±10}{-2}$$

Now solve the equation x=\frac{-70±10}{-2} when ± is plus. Add -70 to 10.

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

Divide -60 by -2.

$$x=30$$

Now solve the equation x=\frac{-70±10}{-2} when ± is minus. Subtract 10 from -70.

$$x=-\frac{80}{-2}$$

Divide -80 by -2.

$$x=40$$

The equation is now solved.

$$x=30 x=40$$

Steps for Completing the Square

Use the distributive property to multiply 100-x by 30+x and combine like terms.

$$3000+70x-x^{2}=4200$$

Subtract 3000 from both sides.

$$70x-x^{2}=4200-3000$$

Subtract 3000 from 4200 to get 1200.

$$70x-x^{2}=1200$$

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.

$$-x^{2}+70x=1200$$

Divide both sides by -1.

$$\frac{-x^{2}+70x}{-1}=\frac{1200}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{70}{-1}x=\frac{1200}{-1}$$

Divide 70 by -1.

$$x^{2}-70x=\frac{1200}{-1}$$

Divide 1200 by -1.

$$x^{2}-70x=-1200$$

Divide -70, the coefficient of the x term, by 2 to get -35. Then add the square of -35 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-70x+\left(-35\right)^{2}=-1200+\left(-35\right)^{2}$$

Square -35.

$$x^{2}-70x+1225=-1200+1225$$

Add -1200 to 1225.

$$x^{2}-70x+1225=25$$

Factor x^{2}-70x+1225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-35\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-35\right)^{2}}=\sqrt{25}$$

Simplify.

$$x-35=5 x-35=-5$$

Add 35 to both sides of the equation.

$$x=40 x=30$$