Solving using the quadratic formula

The quadratic formula is a general method used to solve any quadratic equation of the form $a{x}^2+bx+c=0$, where $a$, $b$, and $c$ are constants. The formula is particularly useful when factoring is not easy or possible. The quadratic formula is:

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

Follow the steps below to solve quadratic equations using the quadratic formula.

1. Write the quadratic equation in standard form:

Ensure the quadratic equation is written as $a{x}^2+bx+c=0$, where $a$, $b$, and $c$ are constants.

2. Identify the values of $a$, $b$, and $c$:

Recognise the values of $a$ (the coefficient of $x^2$), $b$ (the coefficient of $x$), and $c$ (the constant term) in the quadratic equation.

3. Substitute the values of $a$, $b$, and $c$ into the quadratic formula:

Substitute the identified values of $a$, $b$, and $c$ into the formula: $$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

4. Solve for $x$:

The plus-minus symbol ( $\pm$ ) means that there is a solution when it is a plus and when it is a minus. This will normally therefore give you two different answers.

Example

Solve the quadratic equation $2x^2+4x-6=0$ using the quadratic formula.

1. Write the equation in standard form:

The equation is already in standard form: $2x^2+4x-6=0$.

2. Identify $a=2$, $b=4$, and $c=-6$:

The values are $a=2$, $b=4$, and $c=-6$.

3. Substitute into the quadratic formula:

Substitute $a=2$, $b=4$, and $c=-6$ into the quadratic formula: $$x=\frac{-4\pm \sqrt{4^2-4(2)(-6)}}{2(2)}$$

4. Solve for $x$:

There are two possible solutions: $$x_1=\frac{-4+8}{4}=\frac{4}{4}=1$$ and $$x_2=\frac{-4-8}{4}=\frac{-12}{4}=-3$$

The solutions are:

$x=1$ and $x=-3$.