Solve frac{-b+sqrt{b-ac}}{2a}frac{-b-sqrt{b^-ac}}{2a}

$$\frac{ -b+ \sqrt{ b-ac } }{ 2a } \frac{ -b- \sqrt{ { b }^{ } -ac } }{ 2a }$$

Evaluate

$$\frac{ac+b^{2}-b}{4a^{2}}$$

Short Solution Steps

Calculate b to the power of 1 and get b.

$$\frac{-b+\sqrt{b-ac}}{2a}\times \frac{-b-\sqrt{b-ac}}{2a}$$

Multiply \frac{-b+\sqrt{b-ac}}{2a} times \frac{-b-\sqrt{b-ac}}{2a} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(-b+\sqrt{b-ac}\right)\left(-b-\sqrt{b-ac}\right)}{2a\times 2a}$$

Multiply a and a to get a^{2}.

$$\frac{\left(-b+\sqrt{b-ac}\right)\left(-b-\sqrt{b-ac}\right)}{2a^{2}\times 2}$$

Consider \left(-b+\sqrt{b-ac}\right)\left(-b-\sqrt{b-ac}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(-b\right)^{2}-\left(\sqrt{b-ac}\right)^{2}}{2a^{2}\times 2}$$

Calculate -b to the power of 2 and get b^{2}.

$$\frac{b^{2}-\left(\sqrt{b-ac}\right)^{2}}{2a^{2}\times 2}$$

Calculate \sqrt{b-ac} to the power of 2 and get b-ac.

$$\frac{b^{2}-\left(b-ac\right)}{2a^{2}\times 2}$$

Multiply 2 and 2 to get 4.

$$\frac{b^{2}-\left(b-ac\right)}{4a^{2}}$$

To find the opposite of b-ac, find the opposite of each term.

$$\frac{b^{2}-b+ac}{4a^{2}}$$

Factor

$$\frac{ac+b^{2}-b}{4a^{2}}$$