Solve (2+a)a/2=5

$$( 2 + a ) \frac { a } { 2 } = 5$$

Solve for a

$$a=\sqrt{11}-1\approx 2.31662479$$

$$a=-\sqrt{11}-1\approx -4.31662479$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$\left(2+a\right)a=10$$

Use the distributive property to multiply 2+a by a.

$$2a+a^{2}=10$$

Subtract 10 from both sides.

$$2a+a^{2}-10=0$$

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.

$$a^{2}+2a-10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and -10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-2±\sqrt{2^{2}-4\left(-10\right)}}{2}$$

Square 2.

$$a=\frac{-2±\sqrt{4-4\left(-10\right)}}{2}$$

Multiply -4 times -10.

$$a=\frac{-2±\sqrt{4+40}}{2}$$

Add 4 to 40.

$$a=\frac{-2±\sqrt{44}}{2}$$

Take the square root of 44.

$$a=\frac{-2±2\sqrt{11}}{2}$$

Now solve the equation a=\frac{-2±2\sqrt{11}}{2} when ± is plus. Add -2 to 2\sqrt{11}.

$$a=\frac{2\sqrt{11}-2}{2}$$

Divide -2+2\sqrt{11} by 2.

$$a=\sqrt{11}-1$$

Now solve the equation a=\frac{-2±2\sqrt{11}}{2} when ± is minus. Subtract 2\sqrt{11} from -2.

$$a=\frac{-2\sqrt{11}-2}{2}$$

Divide -2-2\sqrt{11} by 2.

$$a=-\sqrt{11}-1$$

The equation is now solved.

$$a=\sqrt{11}-1 a=-\sqrt{11}-1$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$\left(2+a\right)a=10$$

Use the distributive property to multiply 2+a by a.

$$2a+a^{2}=10$$

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.

$$a^{2}+2a=10$$

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+2a+1^{2}=10+1^{2}$$

Square 1.

$$a^{2}+2a+1=10+1$$

Add 10 to 1.

$$a^{2}+2a+1=11$$

Factor a^{2}+2a+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a+1\right)^{2}=11$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+1\right)^{2}}=\sqrt{11}$$

Simplify.

$$a+1=\sqrt{11} a+1=-\sqrt{11}$$

Subtract 1 from both sides of the equation.

$$a=\sqrt{11}-1 a=-\sqrt{11}-1$$