Solve (12-x)(100+20x)=1400

$$(12-x)(100+20x)=1400$$

Solve for x

$$x=2$$

$$x=5$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 12-x by 100+20x and combine like terms.

$$1200+140x-20x^{2}=1400$$

Subtract 1400 from both sides.

$$1200+140x-20x^{2}-1400=0$$

Subtract 1400 from 1200 to get -200.

$$-200+140x-20x^{2}=0$$

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.

$$-20x^{2}+140x-200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -20 for a, 140 for b, and -200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-140±\sqrt{140^{2}-4\left(-20\right)\left(-200\right)}}{2\left(-20\right)}$$

Square 140.

$$x=\frac{-140±\sqrt{19600-4\left(-20\right)\left(-200\right)}}{2\left(-20\right)}$$

Multiply -4 times -20.

$$x=\frac{-140±\sqrt{19600+80\left(-200\right)}}{2\left(-20\right)}$$

Multiply 80 times -200.

$$x=\frac{-140±\sqrt{19600-16000}}{2\left(-20\right)}$$

Add 19600 to -16000.

$$x=\frac{-140±\sqrt{3600}}{2\left(-20\right)}$$

Take the square root of 3600.

$$x=\frac{-140±60}{2\left(-20\right)}$$

Multiply 2 times -20.

$$x=\frac{-140±60}{-40}$$

Now solve the equation x=\frac{-140±60}{-40} when ± is plus. Add -140 to 60.

$$x=-\frac{80}{-40}$$

Divide -80 by -40.

$$x=2$$

Now solve the equation x=\frac{-140±60}{-40} when ± is minus. Subtract 60 from -140.

$$x=-\frac{200}{-40}$$

Divide -200 by -40.

$$x=5$$

The equation is now solved.

$$x=2 x=5$$

Steps for Completing the Square

Use the distributive property to multiply 12-x by 100+20x and combine like terms.

$$1200+140x-20x^{2}=1400$$

Subtract 1200 from both sides.

$$140x-20x^{2}=1400-1200$$

Subtract 1200 from 1400 to get 200.

$$140x-20x^{2}=200$$

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.

$$-20x^{2}+140x=200$$

Divide both sides by -20.

$$\frac{-20x^{2}+140x}{-20}=\frac{200}{-20}$$

Dividing by -20 undoes the multiplication by -20.

$$x^{2}+\frac{140}{-20}x=\frac{200}{-20}$$

Divide 140 by -20.

$$x^{2}-7x=\frac{200}{-20}$$

Divide 200 by -20.

$$x^{2}-7x=-10$$

Divide -7, the coefficient of the x term, by 2 to get -\frac{7}{2}. Then add the square of -\frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-7x+\left(-\frac{7}{2}\right)^{2}=-10+\left(-\frac{7}{2}\right)^{2}$$

Square -\frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-7x+\frac{49}{4}=-10+\frac{49}{4}$$

Add -10 to \frac{49}{4}.

$$x^{2}-7x+\frac{49}{4}=\frac{9}{4}$$

Factor x^{2}-7x+\frac{49}{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}.

$$\left(x-\frac{7}{2}\right)^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Simplify.

$$x-\frac{7}{2}=\frac{3}{2} x-\frac{7}{2}=-\frac{3}{2}$$

Add \frac{7}{2} to both sides of the equation.

$$x=5 x=2$$