Solve (x-40)(-30x+2100)=6270

$$(x-40)(-30x+2100)=6270$$

Solve for x

$$x=51$$

$$x=59$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-40 by -30x+2100 and combine like terms.

$$-30x^{2}+3300x-84000=6270$$

Subtract 6270 from both sides.

$$-30x^{2}+3300x-84000-6270=0$$

Subtract 6270 from -84000 to get -90270.

$$-30x^{2}+3300x-90270=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -30 for a, 3300 for b, and -90270 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3300±\sqrt{3300^{2}-4\left(-30\right)\left(-90270\right)}}{2\left(-30\right)}$$

Square 3300.

$$x=\frac{-3300±\sqrt{10890000-4\left(-30\right)\left(-90270\right)}}{2\left(-30\right)}$$

Multiply -4 times -30.

$$x=\frac{-3300±\sqrt{10890000+120\left(-90270\right)}}{2\left(-30\right)}$$

Multiply 120 times -90270.

$$x=\frac{-3300±\sqrt{10890000-10832400}}{2\left(-30\right)}$$

Add 10890000 to -10832400.

$$x=\frac{-3300±\sqrt{57600}}{2\left(-30\right)}$$

Take the square root of 57600.

$$x=\frac{-3300±240}{2\left(-30\right)}$$

Multiply 2 times -30.

$$x=\frac{-3300±240}{-60}$$

Now solve the equation x=\frac{-3300±240}{-60} when ± is plus. Add -3300 to 240.

$$x=-\frac{3060}{-60}$$

Divide -3060 by -60.

$$x=51$$

Now solve the equation x=\frac{-3300±240}{-60} when ± is minus. Subtract 240 from -3300.

$$x=-\frac{3540}{-60}$$

Divide -3540 by -60.

$$x=59$$

The equation is now solved.

$$x=51 x=59$$

Steps for Completing the Square

Use the distributive property to multiply x-40 by -30x+2100 and combine like terms.

$$-30x^{2}+3300x-84000=6270$$

Add 84000 to both sides.

$$-30x^{2}+3300x=6270+84000$$

Add 6270 and 84000 to get 90270.

$$-30x^{2}+3300x=90270$$

Divide both sides by -30.

$$\frac{-30x^{2}+3300x}{-30}=\frac{90270}{-30}$$

Dividing by -30 undoes the multiplication by -30.

$$x^{2}+\frac{3300}{-30}x=\frac{90270}{-30}$$

Divide 3300 by -30.

$$x^{2}-110x=\frac{90270}{-30}$$

Divide 90270 by -30.

$$x^{2}-110x=-3009$$

Divide -110, the coefficient of the x term, by 2 to get -55. Then add the square of -55 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-110x+\left(-55\right)^{2}=-3009+\left(-55\right)^{2}$$

Square -55.

$$x^{2}-110x+3025=-3009+3025$$

Add -3009 to 3025.

$$x^{2}-110x+3025=16$$

Factor x^{2}-110x+3025. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-55\right)^{2}=16$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-55\right)^{2}}=\sqrt{16}$$

Simplify.

$$x-55=4 x-55=-4$$

Add 55 to both sides of the equation.

$$x=59 x=51$$