Solve x(30-x)/2=2

$$x \frac { ( 30 - x ) } { 2 } = 2$$

Solve for x

$$x=\sqrt{221}+15\approx 29.866068747$$

$$x=15-\sqrt{221}\approx 0.133931253$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$x\left(30-x\right)=4$$

Use the distributive property to multiply x by 30-x.

$$30x-x^{2}=4$$

Subtract 4 from both sides.

$$30x-x^{2}-4=0$$

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.

$$-x^{2}+30x-4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 30 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-30±\sqrt{30^{2}-4\left(-1\right)\left(-4\right)}}{2\left(-1\right)}$$

Square 30.

$$x=\frac{-30±\sqrt{900-4\left(-1\right)\left(-4\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-30±\sqrt{900+4\left(-4\right)}}{2\left(-1\right)}$$

Multiply 4 times -4.

$$x=\frac{-30±\sqrt{900-16}}{2\left(-1\right)}$$

Add 900 to -16.

$$x=\frac{-30±\sqrt{884}}{2\left(-1\right)}$$

Take the square root of 884.

$$x=\frac{-30±2\sqrt{221}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-30±2\sqrt{221}}{-2}$$

Now solve the equation x=\frac{-30±2\sqrt{221}}{-2} when ± is plus. Add -30 to 2\sqrt{221}.

$$x=\frac{2\sqrt{221}-30}{-2}$$

Divide -30+2\sqrt{221} by -2.

$$x=15-\sqrt{221}$$

Now solve the equation x=\frac{-30±2\sqrt{221}}{-2} when ± is minus. Subtract 2\sqrt{221} from -30.

$$x=\frac{-2\sqrt{221}-30}{-2}$$

Divide -30-2\sqrt{221} by -2.

$$x=\sqrt{221}+15$$

The equation is now solved.

$$x=15-\sqrt{221} x=\sqrt{221}+15$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$x\left(30-x\right)=4$$

Use the distributive property to multiply x by 30-x.

$$30x-x^{2}=4$$

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.

$$-x^{2}+30x=4$$

Divide both sides by -1.

$$\frac{-x^{2}+30x}{-1}=\frac{4}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{30}{-1}x=\frac{4}{-1}$$

Divide 30 by -1.

$$x^{2}-30x=\frac{4}{-1}$$

Divide 4 by -1.

$$x^{2}-30x=-4$$

Divide -30, the coefficient of the x term, by 2 to get -15. Then add the square of -15 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-30x+\left(-15\right)^{2}=-4+\left(-15\right)^{2}$$

Square -15.

$$x^{2}-30x+225=-4+225$$

Add -4 to 225.

$$x^{2}-30x+225=221$$

Factor x^{2}-30x+225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-15\right)^{2}=221$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-15\right)^{2}}=\sqrt{221}$$

Simplify.

$$x-15=\sqrt{221} x-15=-\sqrt{221}$$

Add 15 to both sides of the equation.

$$x=\sqrt{221}+15 x=15-\sqrt{221}$$