Solve 4a^2-4a+8>0

$$4 a ^ { 2 } - 4 a + 8 > 0$$

Solve for a

$$a\in \mathrm{R}$$

Steps Using the Quadratic Formula

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.

$$4a^{2}-4a+8=0$$

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, -4 for b, and 8 for c in the quadratic formula.

$$a=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 4\times 8}}{2\times 4}$$

Do the calculations.

$$a=\frac{4±\sqrt{-112}}{8}$$

Since the square root of a negative number is not defined in the real field, there are no solutions. Expression 4a^{2}-4a+8 has the same sign for any a. To determine the sign, calculate the value of the expression for a=0.

$$4\times 0^{2}-4\times 0+8=8$$

The value of the expression 4a^{2}-4a+8 is always positive. Inequality holds for a\in \mathrm{R}.

$$a\in \mathrm{R}$$