Solve 10x+(x+5)(x-4000)=385220

$$10 x + ( x + 5 ) ( x - 4000 ) = 385220$$

Solve for x

$$x = \frac{\sqrt{17501105} + 3985}{2} \approx 4084.216101673$$

$$x=\frac{3985-\sqrt{17501105}}{2}\approx -99.216101673$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x+5 by x-4000 and combine like terms.

$$10x+x^{2}-3995x-20000=385220$$

Combine 10x and -3995x to get -3985x.

$$-3985x+x^{2}-20000=385220$$

Subtract 385220 from both sides.

$$-3985x+x^{2}-20000-385220=0$$

Subtract 385220 from -20000 to get -405220.

$$-3985x+x^{2}-405220=0$$

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.

$$x^{2}-3985x-405220=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -3985 for b, and -405220 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3985\right)±\sqrt{\left(-3985\right)^{2}-4\left(-405220\right)}}{2}$$

Square -3985.

$$x=\frac{-\left(-3985\right)±\sqrt{15880225-4\left(-405220\right)}}{2}$$

Multiply -4 times -405220.

$$x=\frac{-\left(-3985\right)±\sqrt{15880225+1620880}}{2}$$

Add 15880225 to 1620880.

$$x=\frac{-\left(-3985\right)±\sqrt{17501105}}{2}$$

The opposite of -3985 is 3985.

$$x=\frac{3985±\sqrt{17501105}}{2}$$

Now solve the equation x=\frac{3985±\sqrt{17501105}}{2} when ± is plus. Add 3985 to \sqrt{17501105}.

$$x=\frac{\sqrt{17501105}+3985}{2}$$

Now solve the equation x=\frac{3985±\sqrt{17501105}}{2} when ± is minus. Subtract \sqrt{17501105} from 3985.

$$x=\frac{3985-\sqrt{17501105}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{17501105}+3985}{2} x=\frac{3985-\sqrt{17501105}}{2}$$

Steps for Completing the Square

Use the distributive property to multiply x+5 by x-4000 and combine like terms.

$$10x+x^{2}-3995x-20000=385220$$

Combine 10x and -3995x to get -3985x.

$$-3985x+x^{2}-20000=385220$$

Add 20000 to both sides.

$$-3985x+x^{2}=385220+20000$$

Add 385220 and 20000 to get 405220.

$$-3985x+x^{2}=405220$$

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.

$$x^{2}-3985x=405220$$

Divide -3985, the coefficient of the x term, by 2 to get -\frac{3985}{2}. Then add the square of -\frac{3985}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-3985x+\left(-\frac{3985}{2}\right)^{2}=405220+\left(-\frac{3985}{2}\right)^{2}$$

Square -\frac{3985}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-3985x+\frac{15880225}{4}=405220+\frac{15880225}{4}$$

Add 405220 to \frac{15880225}{4}.

$$x^{2}-3985x+\frac{15880225}{4}=\frac{17501105}{4}$$

Factor x^{2}-3985x+\frac{15880225}{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}.

$$\left(x-\frac{3985}{2}\right)^{2}=\frac{17501105}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3985}{2}\right)^{2}}=\sqrt{\frac{17501105}{4}}$$

Simplify.

$$x-\frac{3985}{2}=\frac{\sqrt{17501105}}{2} x-\frac{3985}{2}=-\frac{\sqrt{17501105}}{2}$$

Add \frac{3985}{2} to both sides of the equation.

$$x=\frac{\sqrt{17501105}+3985}{2} x=\frac{3985-\sqrt{17501105}}{2}$$