Solve 5a^2-2a-3<0

$$5 a ^ { 2 } - 2 a - 3 < 0$$

Solve for a

$$a\in \left(-\frac{3}{5},1\right)$$

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.

$$5a^{2}-2a-3=0$$

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 5 for a, -2 for b, and -3 for c in the quadratic formula.

$$a=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 5\left(-3\right)}}{2\times 5}$$

Do the calculations.

$$a=\frac{2±8}{10}$$

Solve the equation a=\frac{2±8}{10} when ± is plus and when ± is minus.

$$a=1 a=-\frac{3}{5}$$

Rewrite the inequality by using the obtained solutions.

$$5\left(a-1\right)\left(a+\frac{3}{5}\right)<0$$

For the product to be negative, a-1 and a+\frac{3}{5} have to be of the opposite signs. Consider the case when a-1 is positive and a+\frac{3}{5} is negative.

$$a-1>0 a+\frac{3}{5}<0$$

This is false for any a.

$$a\in \emptyset$$

Consider the case when a+\frac{3}{5} is positive and a-1 is negative.

$$a+\frac{3}{5}>0 a-1<0$$

The solution satisfying both inequalities is a\in \left(-\frac{3}{5},1\right).

$$a\in \left(-\frac{3}{5},1\right)$$

The final solution is the union of the obtained solutions.

$$a\in \left(-\frac{3}{5},1\right)$$