Solve 5x^2+24x-5<0

$$5 x ^ { 2 } + 24 x - 5 < 0$$

Solve for x

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

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.

$$5x^{2}+24x-5=0$$

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, 24 for b, and -5 for c in the quadratic formula.

$$x=\frac{-24±\sqrt{24^{2}-4\times 5\left(-5\right)}}{2\times 5}$$

Do the calculations.

$$x=\frac{-24±26}{10}$$

Solve the equation x=\frac{-24±26}{10} when ± is plus and when ± is minus.

$$x=\frac{1}{5} x=-5$$

Rewrite the inequality by using the obtained solutions.

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

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

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

This is false for any x.

$$x\in \emptyset$$

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

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

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

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

The final solution is the union of the obtained solutions.

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