Solve (19-x)(300+30x)=6300

$$( 19 - x ) ( 300 + 30 x ) = 6300$$

Solve for x

$$x=4$$

$$x=5$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 19-x by 300+30x and combine like terms.

$$5700+270x-30x^{2}=6300$$

Subtract 6300 from both sides.

$$5700+270x-30x^{2}-6300=0$$

Subtract 6300 from 5700 to get -600.

$$-600+270x-30x^{2}=0$$

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.

$$-30x^{2}+270x-600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -30 for a, 270 for b, and -600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-270±\sqrt{270^{2}-4\left(-30\right)\left(-600\right)}}{2\left(-30\right)}$$

Square 270.

$$x=\frac{-270±\sqrt{72900-4\left(-30\right)\left(-600\right)}}{2\left(-30\right)}$$

Multiply -4 times -30.

$$x=\frac{-270±\sqrt{72900+120\left(-600\right)}}{2\left(-30\right)}$$

Multiply 120 times -600.

$$x=\frac{-270±\sqrt{72900-72000}}{2\left(-30\right)}$$

Add 72900 to -72000.

$$x=\frac{-270±\sqrt{900}}{2\left(-30\right)}$$

Take the square root of 900.

$$x=\frac{-270±30}{2\left(-30\right)}$$

Multiply 2 times -30.

$$x=\frac{-270±30}{-60}$$

Now solve the equation x=\frac{-270±30}{-60} when ± is plus. Add -270 to 30.

$$x=-\frac{240}{-60}$$

Divide -240 by -60.

$$x=4$$

Now solve the equation x=\frac{-270±30}{-60} when ± is minus. Subtract 30 from -270.

$$x=-\frac{300}{-60}$$

Divide -300 by -60.

$$x=5$$

The equation is now solved.

$$x=4 x=5$$

Steps for Completing the Square

Use the distributive property to multiply 19-x by 300+30x and combine like terms.

$$5700+270x-30x^{2}=6300$$

Subtract 5700 from both sides.

$$270x-30x^{2}=6300-5700$$

Subtract 5700 from 6300 to get 600.

$$270x-30x^{2}=600$$

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.

$$-30x^{2}+270x=600$$

Divide both sides by -30.

$$\frac{-30x^{2}+270x}{-30}=\frac{600}{-30}$$

Dividing by -30 undoes the multiplication by -30.

$$x^{2}+\frac{270}{-30}x=\frac{600}{-30}$$

Divide 270 by -30.

$$x^{2}-9x=\frac{600}{-30}$$

Divide 600 by -30.

$$x^{2}-9x=-20$$

Divide -9, the coefficient of the x term, by 2 to get -\frac{9}{2}. Then add the square of -\frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=-20+\left(-\frac{9}{2}\right)^{2}$$

Square -\frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-9x+\frac{81}{4}=-20+\frac{81}{4}$$

Add -20 to \frac{81}{4}.

$$x^{2}-9x+\frac{81}{4}=\frac{1}{4}$$

Factor x^{2}-9x+\frac{81}{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}.

$$\left(x-\frac{9}{2}\right)^{2}=\frac{1}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{\frac{1}{4}}$$

Simplify.

$$x-\frac{9}{2}=\frac{1}{2} x-\frac{9}{2}=-\frac{1}{2}$$

Add \frac{9}{2} to both sides of the equation.

$$x=5 x=4$$