Megoldás - 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:

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

$$x=\left(-1*-6\pm sqrt\left(-6^2-4*-1*-9\right)\right)/\left(2*-1\right)$$

$$x=\left(-1*-6\pm sqrt\left(36-4*-1*-9\right)\right)/\left(2*-1\right)$$

$$x=\left(-1*-6\pm sqrt\left(36--4*-9\right)\right)/\left(2*-1\right)$$

$$x=\left(-1*-6\pm sqrt\left(36-36\right)\right)/\left(2*-1\right)$$

$$x=\left(-1*-6\pm sqrt\left(0\right)\right)/\left(2*-1\right)$$

$$x=\left(-1*-6\pm sqrt\left(0\right)\right)/\left(-2\right)$$

$$x=\left(6\pm sqrt\left(0\right)\right)/\left(-2\right)$$

to get the result:

$$x=\left(6\pm sqrt\left(0\right)\right)/\left(-2\right)$$

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

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

$$x=\left(6\pm 0\right)/\left(-2\right)$$

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

Separate the equations:
$$x_1=\left(6+0\right)/\left(-2\right)$$ and $$x_2=\left(6-0\right)/\left(-2\right)$$

$$x_1=\left(6+0\right)/\left(-2\right)$$

$$x_1=\left(6+0\right)/\left(-2\right)$$

$$x_1=\left(6\right)/\left(-2\right)$$

$$x_1=\frac{6}{-}2$$

$$x_1=-3$$

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: -3.

Since the $$a$$ coefficient is negative ($$a$$=-1), this is a "negative" quadratic inequality and the parabola points downward, like a frown.

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 $$-1x^2-6x-9<0$$ has a $$<$$ inequality sign, we look for the parabola intervals that are below the x-axis.

Solution: $$$$

Interval notation: $$$$