## Quadratic Inequalities

For the parabola

on the *x*-axis

above the *x*-axis

below the *x*-axis

### Examples:

1. Solve .

### Show Answer

For *x*-intercepts, *y=0*. *a*>0 so it is concave upwards.

on and above the *x*-axis.

So on and above the *x-*axis.

2. Solve .

### Show Answer

For *x*-intercepts, *y=0. **a<0* so it is concave downwards.

above the *x*-axis.

So above the *x-*axis.

## The Discriminant

The values of *x* and *y* that satisfy a quadratic equation are called the **roots** of the equation. The roots of are the *x*-intercepts of the graph .

In the quadratic formula . the expression is called the **discriminant**. It gives us information about the roots of the quadratic equation .

Notice that when there are 2 real roots the discriminant .

When there are 2 equal roots (or just 1 real root), .

When there are no real roots, .

We often use .

### Examples:

1. Show that the equation has no real roots.

### Show Answer

So the equation has no real roots.

2. Find the values of *k* for which the quadratic equation has real roots.

### Show Answer

3. Show that for all *x*.

### Show Answer

## Quadratic Identities

As a general rule: if two quadratic expressions are equivalent to each other then the corresponding coefficients must be equal.

If for all real *x* then

### Examples:

1. Find values for *a*, *b* and *c* if .

### Show Answer

2. Find the equation of the parabola that passes through the points (-1, -3), (0, 3) and (2, 21).

### Show Answer

## Sum and Product of Roots

The general quadratic equation can be written in the for

where and are the roots of the equation.

If and are the roots of the quadratic equation ,

Sum of roots:

Product of roots:

### Examples

1. Find the quadratic equation that has roots 6 and -1.

### Show Answer

Using the general formula ,

Therefore, the quadratic equation is:

2. Find the quadratic equation that has roots and .

### Show Answer

Using the general formula ,

Therefore, the quadratic equation is:

3. Find the value of *k* if one root of is -2.

### Show Answer

4. Evaluate *p* if one root of is double the other root.

### Show Answer

If one root is then the other root is .

Sum of roots:

Product of roots: