Solve xleft(30-xright)/2=112

$$\frac{ x \left( 30-x \right) }{ 2 } = 112$$

Solve for x

$$x=14$$

$$x=16$$

Steps Using the Quadratic Formula

Multiply both sides by 2.

$$x\left(30-x\right)=112\times 2$$

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

$$30x-x^{2}=112\times 2$$

Multiply 112 and 2 to get 224.

$$30x-x^{2}=224$$

Subtract 224 from both sides.

$$30x-x^{2}-224=0$$

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.

$$-x^{2}+30x-224=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 30 for b, and -224 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-30±\sqrt{30^{2}-4\left(-1\right)\left(-224\right)}}{2\left(-1\right)}$$

Square 30.

$$x=\frac{-30±\sqrt{900-4\left(-1\right)\left(-224\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-30±\sqrt{900+4\left(-224\right)}}{2\left(-1\right)}$$

Multiply 4 times -224.

$$x=\frac{-30±\sqrt{900-896}}{2\left(-1\right)}$$

Add 900 to -896.

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

Take the square root of 4.

$$x=\frac{-30±2}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-30±2}{-2}$$

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

$$x=-\frac{28}{-2}$$

Divide -28 by -2.

$$x=14$$

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

$$x=-\frac{32}{-2}$$

Divide -32 by -2.

$$x=16$$

The equation is now solved.

$$x=14 x=16$$

Steps for Completing the Square

Multiply both sides by 2.

$$x\left(30-x\right)=112\times 2$$

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

$$30x-x^{2}=112\times 2$$

Multiply 112 and 2 to get 224.

$$30x-x^{2}=224$$

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.

$$-x^{2}+30x=224$$

Divide both sides by -1.

$$\frac{-x^{2}+30x}{-1}=\frac{224}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{30}{-1}x=\frac{224}{-1}$$

Divide 30 by -1.

$$x^{2}-30x=\frac{224}{-1}$$

Divide 224 by -1.

$$x^{2}-30x=-224$$

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.

$$x^{2}-30x+\left(-15\right)^{2}=-224+\left(-15\right)^{2}$$

Square -15.

$$x^{2}-30x+225=-224+225$$

Add -224 to 225.

$$x^{2}-30x+225=1$$

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}.

$$\left(x-15\right)^{2}=1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-15\right)^{2}}=\sqrt{1}$$

Simplify.

$$x-15=1 x-15=-1$$

Add 15 to both sides of the equation.

$$x=16 x=14$$