Solve b^2+b+1/4=0

$$b ^ { 2 } + b + \frac { 1 } { 4 } = 0$$

Solve for b

$$b=-\frac{1}{2}=-0.5$$

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.

$$b^{2}+b+\frac{1}{4}=0$$

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

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

Square 1.

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

Multiply -4 times \frac{1}{4}.

$$b=\frac{-1±\sqrt{1-1}}{2}$$

Add 1 to -1.

$$b=\frac{-1±\sqrt{0}}{2}$$

Take the square root of 0.

$$b=-\frac{1}{2}$$

Steps for Completing the Square

Factor b^{2}+b+\frac{1}{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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$b+\frac{1}{2}=0 b+\frac{1}{2}=0$$

Subtract \frac{1}{2} from both sides of the equation.

$$b=-\frac{1}{2} b=-\frac{1}{2}$$

The equation is now solved. Solutions are the same.

$$b=-\frac{1}{2}$$