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:

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

$$x=\left(-12\pm sqrt\left({12}^2-4*9*4\right)\right)/\left(2*9\right)$$

$$x=\left(-12\pm sqrt\left(144-4*9*4\right)\right)/\left(2*9\right)$$

$$x=\left(-12\pm sqrt\left(144-36*4\right)\right)/\left(2*9\right)$$

$$x=\left(-12\pm sqrt\left(144-144\right)\right)/\left(2*9\right)$$

$$x=\left(-12\pm sqrt\left(0\right)\right)/\left(2*9\right)$$

$$x=\left(-12\pm sqrt\left(0\right)\right)/\left(18\right)$$

to get the result:

$$x=\left(-12\pm sqrt\left(0\right)\right)/18$$

Simplify $$0$$ by finding its prime factors:

The prime factorization of $$0$$ is $$0$$

$$x=\left(-12\pm 0\right)/18$$

The ± means two roots are possible, but since zero is result of the square root, we have one root:

Separate the equations:
$$x_1=\left(-12+0\right)/18$$ and $$x_2=\left(-12-0\right)/18$$

$$x_1=\left(-12+0\right)/18$$

$$x_1=\left(-12+0\right)/18$$

$$x_1=\left(-12\right)/18$$

$$x_1=-\frac{12}{18}$$

$$x_1=-0.667$$

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: -0.667.

Since the $$a$$ coefficient is positive ($$a$$=9), 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 $$9x^2+12x+4\le 0$$ has a $$\le$$ inequality sign, we look for the parabola intervals that are below the x-axis.

Solution: $$$$

Interval notation: $$$$