Solve -x^2/2+x+1/2

$$- \frac{ { x }^{ 2 } }{ 2 } +x+ \frac{ 1 }{ 2 }$$

Factor

$$-\frac{1}{2}\left(x-\left(1-\sqrt{2}\right)\right)\left(x-\left(\sqrt{2}+1\right)\right)$$

Solution Steps

Since \frac{x^{2}}{2} and \frac{1}{2} have the same denominator, add them by adding their numerators.

$$factor(\frac{x^{2}+1}{2}+x)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{2}{2}.

$$factor(\frac{x^{2}+1}{2}+\frac{2x}{2})$$

Since \frac{x^{2}+1}{2} and \frac{2x}{2} have the same denominator, add them by adding their numerators.

$$factor(\frac{x^{2}+1+2x}{2})$$

Consider x^{2}+1+2x. Use the perfect square formula, a^{2}+2ab+b^{2}=\left(a+b\right)^{2}, where a=x and b=1.

$$\left(x+1\right)^{2}$$

Rewrite the complete factored expression.

$$\frac{\left(x+1\right)^{2}}{2}$$

Evaluate

$$-\frac{x^{2}}{2}+x+\frac{1}{2}$$

Short Solution Steps

Since \frac{x^{2}}{2} and \frac{1}{2} have the same denominator, add them by adding their numerators.

$$\frac{x^{2}+1}{2}+x$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{2}{2}.

$$\frac{x^{2}+1}{2}+\frac{2x}{2}$$

Since \frac{x^{2}+1}{2} and \frac{2x}{2} have the same denominator, add them by adding their numerators.

$$\frac{x^{2}+1+2x}{2}$$