Quadratic Equations

A quadratic equation is an equation where the highest power of the variable is 2. It can be written in the form $ax^2+bx+c=0$, where $a$ does not equal 0.

$a$ is the coefficient of $x^2$, not equal to 0

$b$ is the coefficient of $x$

$c$ is the constant term

For example, in the equation $2x^2-3x+5=0$, the values are $a=2$, $b=-3$, and $c=5$.

Quadratic equations can have 0, 1, or 2 solutions, depending on the equation. Consider these examples:

1. $x^2=-4$ has 0 solutions because a squared number cannot be negative.

2. $x^2=0$ has 1 solution: $x=0$.

3. $x^2=9$ has 2 solutions: $x=3$ and $x=-3$, or $x=\pm3$.

The symbol ± means “plus or minus,” representing both the positive and negative square roots.

Equations from Graphs

To find the equation of a quadratic, we can follow these steps:

1. Identify the $x$-values of the $x$-intercepts. These occur at the point where $y=0$.

2. Substitute the $x$-values found in Step 1 for $p$ and $q$ in the equation $y=a\left(x-p\right)\left(x-q\right)$.

3. Find the value of $a$ by using another known point or by using second differences.

Further Quadratic Equations

We have looked at a number of ways to solve quadratic equations. These include:

• Factorising the quadratic expression and then using the null factor law. In particular, when the quadratic expression is non-monic, we can use techniques for non-monic quadratic trinomials.

• Completing the square and then taking the square root of both sides of the equation.

• Using the quadratic formula.