Solution - Solving quadratic inequalities using the quadratic formula

To find the roots of a quadratic equation, plug its coefficients ($$a$$, $$b$$ and $$c$$ ) into the quadratic formula:

$$m=\left(-b\pm sqrt\left(b^2-4ac\right)\right)/\left(2a\right)$$

$$m=\left(-1*-5\pm sqrt\left(-5^2-4*4*-5\right)\right)/\left(2*4\right)$$

$$m=\left(-1*-5\pm sqrt\left(25-4*4*-5\right)\right)/\left(2*4\right)$$

$$m=\left(-1*-5\pm sqrt\left(25-16*-5\right)\right)/\left(2*4\right)$$

$$m=\left(-1*-5\pm sqrt\left(25--80\right)\right)/\left(2*4\right)$$

$$m=\left(-1*-5\pm sqrt\left(25+80\right)\right)/\left(2*4\right)$$

$$m=\left(-1*-5\pm sqrt\left(105\right)\right)/\left(2*4\right)$$

$$m=\left(-1*-5\pm sqrt\left(105\right)\right)/\left(8\right)$$

$$m=\left(5\pm sqrt\left(105\right)\right)/8$$

to get the result:

$$m=\left(5\pm sqrt\left(105\right)\right)/8$$

Simplify $$105$$ by finding its prime factors:

The prime factorization of $$105$$ is $$3\cdot 5\cdot 7$$

$$m=\left(5\pm sqrt\left(105\right)\right)/8$$

The ± means two roots are possible.

Separate the equations:
$$m_1=\left(5+sqrt\left(105\right)\right)/8$$ and $$m_2=\left(5-sqrt\left(105\right)\right)/8$$

$$m_1=\left(5+sqrt\left(105\right)\right)/8$$

$$m_1=\left(5+sqrt\left(105\right)\right)/8$$

$$m_1=\left(5+10.247\right)/8$$

$$m_1=\left(5+10.247\right)/8$$

$$m_1=\left(15.247\right)/8$$

$$m_1=\frac{15.247}{8}$$

$$m_1=1.906$$

$$m_2=\left(5-sqrt\left(105\right)\right)/8$$

$$m_2=\left(5-10.247\right)/8$$

$$m_2=\left(5-10.247\right)/8$$

$$m_2=\left(-5.247\right)/8$$

$$m_2=-\frac{5.247}{8}$$

$$m_2=-0.656$$

To find the intervals of a quadratic inequality, we start by finding its parabola.

The roots of the parabola (where it meets the x-axis) are: -0.656, 1.906.

Since the $$a$$ coefficient is positive ($$a$$=4), this is a "positive" quadratic inequality and the parabola points upward, like a smile!

If the inequality sign is ≤ or ≥ , then the intervals include the roots and we use a solid line. If the inequality sign is < or > the intervals do not include the roots and we use a dotted line.

Since $$4m^2-5m-5>0$$ has a $$>$$ inequality sign, we look for the parabola intervals that are above the x-axis.

Solution: $$$$

Interval notation: $$$$