Solving using the quadratic formula

Topic summary

The quadratic formula is a general method used to solve any quadratic equation of the form \( ax^2 + bx + c = 0 \), 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 \( ax^2 + bx + c = 0 \), 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 \( x = -3 \).