Solve (x-40)(-10x+740)=2250

$$(x-40)(-10x+740)=2250$$

Solve for x

$$x=49$$

$$x=65$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-40 by -10x+740 and combine like terms.

$$-10x^{2}+1140x-29600=2250$$

Subtract 2250 from both sides.

$$-10x^{2}+1140x-29600-2250=0$$

Subtract 2250 from -29600 to get -31850.

$$-10x^{2}+1140x-31850=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 1140 for b, and -31850 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1140±\sqrt{1140^{2}-4\left(-10\right)\left(-31850\right)}}{2\left(-10\right)}$$

Square 1140.

$$x=\frac{-1140±\sqrt{1299600-4\left(-10\right)\left(-31850\right)}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-1140±\sqrt{1299600+40\left(-31850\right)}}{2\left(-10\right)}$$

Multiply 40 times -31850.

$$x=\frac{-1140±\sqrt{1299600-1274000}}{2\left(-10\right)}$$

Add 1299600 to -1274000.

$$x=\frac{-1140±\sqrt{25600}}{2\left(-10\right)}$$

Take the square root of 25600.

$$x=\frac{-1140±160}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{-1140±160}{-20}$$

Now solve the equation x=\frac{-1140±160}{-20} when ± is plus. Add -1140 to 160.

$$x=-\frac{980}{-20}$$

Divide -980 by -20.

$$x=49$$

Now solve the equation x=\frac{-1140±160}{-20} when ± is minus. Subtract 160 from -1140.

$$x=-\frac{1300}{-20}$$

Divide -1300 by -20.

$$x=65$$

The equation is now solved.

$$x=49 x=65$$

Steps for Completing the Square

Use the distributive property to multiply x-40 by -10x+740 and combine like terms.

$$-10x^{2}+1140x-29600=2250$$

Add 29600 to both sides.

$$-10x^{2}+1140x=2250+29600$$

Add 2250 and 29600 to get 31850.

$$-10x^{2}+1140x=31850$$

Divide both sides by -10.

$$\frac{-10x^{2}+1140x}{-10}=\frac{31850}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\frac{1140}{-10}x=\frac{31850}{-10}$$

Divide 1140 by -10.

$$x^{2}-114x=\frac{31850}{-10}$$

Divide 31850 by -10.

$$x^{2}-114x=-3185$$

Divide -114, the coefficient of the x term, by 2 to get -57. Then add the square of -57 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-114x+\left(-57\right)^{2}=-3185+\left(-57\right)^{2}$$

Square -57.

$$x^{2}-114x+3249=-3185+3249$$

Add -3185 to 3249.

$$x^{2}-114x+3249=64$$

Factor x^{2}-114x+3249. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-57\right)^{2}=64$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-57\right)^{2}}=\sqrt{64}$$

Simplify.

$$x-57=8 x-57=-8$$

Add 57 to both sides of the equation.

$$x=65 x=49$$