Solve x*(40-x)=381

$$x \times ( 40 - x ) = 381$$

Solve for x

$$x=\sqrt{19}+20\approx 24.358898944$$

$$x=20-\sqrt{19}\approx 15.641101056$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x by 40-x.

$$40x-x^{2}=381$$

Subtract 381 from both sides.

$$40x-x^{2}-381=0$$

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.

$$-x^{2}+40x-381=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 40 for b, and -381 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-40±\sqrt{40^{2}-4\left(-1\right)\left(-381\right)}}{2\left(-1\right)}$$

Square 40.

$$x=\frac{-40±\sqrt{1600-4\left(-1\right)\left(-381\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-40±\sqrt{1600+4\left(-381\right)}}{2\left(-1\right)}$$

Multiply 4 times -381.

$$x=\frac{-40±\sqrt{1600-1524}}{2\left(-1\right)}$$

Add 1600 to -1524.

$$x=\frac{-40±\sqrt{76}}{2\left(-1\right)}$$

Take the square root of 76.

$$x=\frac{-40±2\sqrt{19}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-40±2\sqrt{19}}{-2}$$

Now solve the equation x=\frac{-40±2\sqrt{19}}{-2} when ± is plus. Add -40 to 2\sqrt{19}.

$$x=\frac{2\sqrt{19}-40}{-2}$$

Divide -40+2\sqrt{19} by -2.

$$x=20-\sqrt{19}$$

Now solve the equation x=\frac{-40±2\sqrt{19}}{-2} when ± is minus. Subtract 2\sqrt{19} from -40.

$$x=\frac{-2\sqrt{19}-40}{-2}$$

Divide -40-2\sqrt{19} by -2.

$$x=\sqrt{19}+20$$

The equation is now solved.

$$x=20-\sqrt{19} x=\sqrt{19}+20$$

Steps for Completing the Square

Use the distributive property to multiply x by 40-x.

$$40x-x^{2}=381$$

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.

$$-x^{2}+40x=381$$

Divide both sides by -1.

$$\frac{-x^{2}+40x}{-1}=\frac{381}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{40}{-1}x=\frac{381}{-1}$$

Divide 40 by -1.

$$x^{2}-40x=\frac{381}{-1}$$

Divide 381 by -1.

$$x^{2}-40x=-381$$

Divide -40, the coefficient of the x term, by 2 to get -20. Then add the square of -20 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-40x+\left(-20\right)^{2}=-381+\left(-20\right)^{2}$$

Square -20.

$$x^{2}-40x+400=-381+400$$

Add -381 to 400.

$$x^{2}-40x+400=19$$

Factor x^{2}-40x+400. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-20\right)^{2}=19$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-20\right)^{2}}=\sqrt{19}$$

Simplify.

$$x-20=\sqrt{19} x-20=-\sqrt{19}$$

Add 20 to both sides of the equation.

$$x=\sqrt{19}+20 x=20-\sqrt{19}$$