Solve (x-40)(-4x+360)=2400

$$(x-40)(-4x+360)=2400$$

Solve for x

$$x=60$$

$$x=70$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-40 by -4x+360 and combine like terms.

$$-4x^{2}+520x-14400=2400$$

Subtract 2400 from both sides.

$$-4x^{2}+520x-14400-2400=0$$

Subtract 2400 from -14400 to get -16800.

$$-4x^{2}+520x-16800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 520 for b, and -16800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-520±\sqrt{520^{2}-4\left(-4\right)\left(-16800\right)}}{2\left(-4\right)}$$

Square 520.

$$x=\frac{-520±\sqrt{270400-4\left(-4\right)\left(-16800\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-520±\sqrt{270400+16\left(-16800\right)}}{2\left(-4\right)}$$

Multiply 16 times -16800.

$$x=\frac{-520±\sqrt{270400-268800}}{2\left(-4\right)}$$

Add 270400 to -268800.

$$x=\frac{-520±\sqrt{1600}}{2\left(-4\right)}$$

Take the square root of 1600.

$$x=\frac{-520±40}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-520±40}{-8}$$

Now solve the equation x=\frac{-520±40}{-8} when ± is plus. Add -520 to 40.

$$x=-\frac{480}{-8}$$

Divide -480 by -8.

$$x=60$$

Now solve the equation x=\frac{-520±40}{-8} when ± is minus. Subtract 40 from -520.

$$x=-\frac{560}{-8}$$

Divide -560 by -8.

$$x=70$$

The equation is now solved.

$$x=60 x=70$$

Steps for Completing the Square

Use the distributive property to multiply x-40 by -4x+360 and combine like terms.

$$-4x^{2}+520x-14400=2400$$

Add 14400 to both sides.

$$-4x^{2}+520x=2400+14400$$

Add 2400 and 14400 to get 16800.

$$-4x^{2}+520x=16800$$

Divide both sides by -4.

$$\frac{-4x^{2}+520x}{-4}=\frac{16800}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\frac{520}{-4}x=\frac{16800}{-4}$$

Divide 520 by -4.

$$x^{2}-130x=\frac{16800}{-4}$$

Divide 16800 by -4.

$$x^{2}-130x=-4200$$

Divide -130, the coefficient of the x term, by 2 to get -65. Then add the square of -65 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-130x+\left(-65\right)^{2}=-4200+\left(-65\right)^{2}$$

Square -65.

$$x^{2}-130x+4225=-4200+4225$$

Add -4200 to 4225.

$$x^{2}-130x+4225=25$$

Factor x^{2}-130x+4225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-65\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-65\right)^{2}}=\sqrt{25}$$

Simplify.

$$x-65=5 x-65=-5$$

Add 65 to both sides of the equation.

$$x=70 x=60$$