Solve 3a+(a^2+1)=1

$$3 a + ( a ^ { 2 } + 1 ) = 1$$

Solve for a

$$a=-3$$

$$a=0$$

Steps Using Factoring

Subtract 1 from both sides.

$$3a+a^{2}+1-1=0$$

Subtract 1 from 1 to get 0.

$$3a+a^{2}=0$$

Factor out a.

$$a\left(3+a\right)=0$$

To find equation solutions, solve a=0 and 3+a=0.

$$a=0 a=-3$$

Steps Using the Quadratic Formula

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.

$$a^{2}+3a+1=1$$

Subtract 1 from both sides of the equation.

$$a^{2}+3a+1-1=1-1$$

Subtracting 1 from itself leaves 0.

$$a^{2}+3a+1-1=0$$

Subtract 1 from 1.

$$a^{2}+3a=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 3 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-3±\sqrt{3^{2}}}{2}$$

Take the square root of 3^{2}.

$$a=\frac{-3±3}{2}$$

Now solve the equation a=\frac{-3±3}{2} when ± is plus. Add -3 to 3.

$$a=\frac{0}{2}$$

Divide 0 by 2.

$$a=0$$

Now solve the equation a=\frac{-3±3}{2} when ± is minus. Subtract 3 from -3.

$$a=-\frac{6}{2}$$

Divide -6 by 2.

$$a=-3$$

The equation is now solved.

$$a=0 a=-3$$

Steps for Completing the Square

Subtract 1 from both sides.

$$3a+a^{2}+1-1=0$$

Subtract 1 from 1 to get 0.

$$3a+a^{2}=0$$

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.

$$a^{2}+3a=0$$

Divide 3, the coefficient of the x term, by 2 to get \frac{3}{2}. Then add the square of \frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+3a+\left(\frac{3}{2}\right)^{2}=\left(\frac{3}{2}\right)^{2}$$

Square \frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$a^{2}+3a+\frac{9}{4}=\frac{9}{4}$$

Factor a^{2}+3a+\frac{9}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a+\frac{3}{2}\right)^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+\frac{3}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Simplify.

$$a+\frac{3}{2}=\frac{3}{2} a+\frac{3}{2}=-\frac{3}{2}$$

Subtract \frac{3}{2} from both sides of the equation.

$$a=0 a=-3$$