Solve (2a+1/2)^2>0

$$( 2 a + \frac { 1 } { 2 } ) ^ { 2 } > 0$$

Solve for a

$$a\neq -\frac{1}{4}$$

Solution Steps

To solve the inequality, factor the left hand side. Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$\left(2a+\frac{1}{2}\right)^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 4 for a, 2 for b, and \frac{1}{4} for c in the quadratic formula.

$$a=\frac{-2±\sqrt{2^{2}-4\times 4\times \frac{1}{4}}}{2\times 4}$$

Do the calculations.

$$a=\frac{-2±0}{8}$$

Solutions are the same.

$$a=-\frac{1}{4}$$

Rewrite the inequality by using the obtained solutions.

$$4\left(a+\frac{1}{4}\right)^{2}>0$$

Inequality holds for a\neq -\frac{1}{4}.

$$a\neq -\frac{1}{4}$$