Solve 25(1+2a+a^2)=14

$$25 ( 1 + 2 a + a ^ { 2 } ) = 14$$

Solve for a

$$a=\frac{\sqrt{14}}{5}-1\approx -0.251668523$$

$$a=-\frac{\sqrt{14}}{5}-1\approx -1.748331477$$

Steps Using the Quadratic Formula

Divide both sides by 25.

$$1+2a+a^{2}=\frac{14}{25}$$

Subtract \frac{14}{25} from both sides.

$$1+2a+a^{2}-\frac{14}{25}=0$$

Subtract \frac{14}{25} from 1 to get \frac{11}{25}.

$$\frac{11}{25}+2a+a^{2}=0$$

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.

$$a^{2}+2a+\frac{11}{25}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and \frac{11}{25} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-2±\sqrt{2^{2}-4\times \frac{11}{25}}}{2}$$

Square 2.

$$a=\frac{-2±\sqrt{4-4\times \frac{11}{25}}}{2}$$

Multiply -4 times \frac{11}{25}.

$$a=\frac{-2±\sqrt{4-\frac{44}{25}}}{2}$$

Add 4 to -\frac{44}{25}.

$$a=\frac{-2±\sqrt{\frac{56}{25}}}{2}$$

Take the square root of \frac{56}{25}.

$$a=\frac{-2±\frac{2\sqrt{14}}{5}}{2}$$

Now solve the equation a=\frac{-2±\frac{2\sqrt{14}}{5}}{2} when ± is plus. Add -2 to \frac{2\sqrt{14}}{5}.

$$a=\frac{\frac{2\sqrt{14}}{5}-2}{2}$$

Divide -2+\frac{2\sqrt{14}}{5} by 2.

$$a=\frac{\sqrt{14}}{5}-1$$

Now solve the equation a=\frac{-2±\frac{2\sqrt{14}}{5}}{2} when ± is minus. Subtract \frac{2\sqrt{14}}{5} from -2.

$$a=\frac{-\frac{2\sqrt{14}}{5}-2}{2}$$

Divide -2-\frac{2\sqrt{14}}{5} by 2.

$$a=-\frac{\sqrt{14}}{5}-1$$

The equation is now solved.

$$a=\frac{\sqrt{14}}{5}-1 a=-\frac{\sqrt{14}}{5}-1$$

Steps for Completing the Square

Divide both sides by 25.

$$1+2a+a^{2}=\frac{14}{25}$$

Subtract 1 from both sides.

$$2a+a^{2}=\frac{14}{25}-1$$

Subtract 1 from \frac{14}{25} to get -\frac{11}{25}.

$$2a+a^{2}=-\frac{11}{25}$$

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.

$$a^{2}+2a=-\frac{11}{25}$$

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.

$$a^{2}+2a+1^{2}=-\frac{11}{25}+1^{2}$$

Square 1.

$$a^{2}+2a+1=-\frac{11}{25}+1$$

Add -\frac{11}{25} to 1.

$$a^{2}+2a+1=\frac{14}{25}$$

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}.

$$\left(a+1\right)^{2}=\frac{14}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+1\right)^{2}}=\sqrt{\frac{14}{25}}$$

Simplify.

$$a+1=\frac{\sqrt{14}}{5} a+1=-\frac{\sqrt{14}}{5}$$

Subtract 1 from both sides of the equation.

$$a=\frac{\sqrt{14}}{5}-1 a=-\frac{\sqrt{14}}{5}-1$$