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:

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

$$s=\left(-1*-14\pm sqrt\left(-{14}^2-4*1*40\right)\right)/\left(2*1\right)$$

$$s=\left(-1*-14\pm sqrt\left(196-4*1*40\right)\right)/\left(2*1\right)$$

$$s=\left(-1*-14\pm sqrt\left(196-4*40\right)\right)/\left(2*1\right)$$

$$s=\left(-1*-14\pm sqrt\left(196-160\right)\right)/\left(2*1\right)$$

$$s=\left(-1*-14\pm sqrt\left(36\right)\right)/\left(2*1\right)$$

$$s=\left(-1*-14\pm sqrt\left(36\right)\right)/\left(2\right)$$

$$s=\left(14\pm sqrt\left(36\right)\right)/2$$

to get the result:

$$s=\left(14\pm sqrt\left(36\right)\right)/2$$

Simplify $$36$$ by finding its prime factors:

The prime factorization of $$36$$ is $$2^2\cdot 3^2$$

$$s=\left(14\pm 6\right)/2$$

The ± means two roots are possible.

Separate the equations:
$$s_1=\left(14+6\right)/2$$ and $$s_2=\left(14-6\right)/2$$

$$s_1=\left(14+6\right)/2$$

$$s_1=\left(14+6\right)/2$$

$$s_1=\left(20\right)/2$$

$$s_1=\frac{20}{2}$$

$$s_1=10$$

$$s_2=\left(14-6\right)/2$$

$$s_2=\left(14-6\right)/2$$

$$s_2=\left(8\right)/2$$

$$s_2=\frac{8}{2}$$

$$s_2=4$$

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: 4, 10.

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 $$s^2-14s+40<0$$ has a $$<$$ inequality sign, we look for the parabola intervals that are below the x-axis.

Solution: $$$$

Interval notation: $$$$