Solution - Solving quadratic inequalities using the quadratic formula

The coefficients of our inequality, $$m^2-7m-10\le 0$$, are:

$$a$$ = 1

$$b$$ = -7

$$c$$ = -10

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

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

$$m=\left(-1*-7\pm sqrt\left(-7^2-4*1*-10\right)\right)/\left(2*1\right)$$

$$m=\left(-1*-7\pm sqrt\left(49-4*1*-10\right)\right)/\left(2*1\right)$$

$$m=\left(-1*-7\pm sqrt\left(49-4*-10\right)\right)/\left(2*1\right)$$

$$m=\left(-1*-7\pm sqrt\left(49--40\right)\right)/\left(2*1\right)$$

$$m=\left(-1*-7\pm sqrt\left(49+40\right)\right)/\left(2*1\right)$$

$$m=\left(-1*-7\pm sqrt\left(89\right)\right)/\left(2*1\right)$$

$$m=\left(-1*-7\pm sqrt\left(89\right)\right)/\left(2\right)$$

$$m=\left(7\pm sqrt\left(89\right)\right)/2$$

to get the result:

$$m=\left(7\pm sqrt\left(89\right)\right)/2$$

Simplify $$89$$ by finding its prime factors:

The prime factorization of $$89$$ is $$89$$

$$m=\left(7\pm sqrt\left(89\right)\right)/2$$

The ± means two roots are possible.

Separate the equations:
$$m_1=\left(7+sqrt\left(89\right)\right)/2$$ and $$m_2=\left(7-sqrt\left(89\right)\right)/2$$

$$m_1=\left(7+sqrt\left(89\right)\right)/2$$

$$m_1=\left(7+sqrt\left(89\right)\right)/2$$

$$m_1=\left(7+9.434\right)/2$$

$$m_1=\left(7+9.434\right)/2$$

$$m_1=\left(16.434\right)/2$$

$$m_1=\frac{16.434}{2}$$

$$m_1=8.217$$

$$m_2=\left(7-sqrt\left(89\right)\right)/2$$

$$m_2=\left(7-9.434\right)/2$$

$$m_2=\left(7-9.434\right)/2$$

$$m_2=\left(-2.434\right)/2$$

$$m_2=-\frac{2.434}{2}$$

$$m_2=-1.217$$

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: -1.217, 8.217.

Since the $$a$$ coefficient is positive ($$a$$=1), 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 $$m^2-7m-10\le 0$$ has a $$\le$$ inequality sign, we look for the parabola intervals that are below the x-axis.

Solution: $$$$

Interval notation: $$$$