Solving by quadratic formula - Higher

Using the quadratic formula is another method of solving quadratic equations that will not factorise. You will need to learn this formula, as well as understanding how to use it.

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The quadratic formula for a quadratic equation in the form of ax^2 + bx + c = 0 is x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

Example

Solve x^2 + 6x - 12 = 0 using the quadratic formula.

First, identify the value of a, b and c. In this example, a = 1, b = 6 and c = -12.

Substitute these values into the quadratic formula:

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

x = \frac{- 6 \pm \sqrt{6^2 - 4 \times 1 \times -12}}{2 \times 1}

x = \frac{-6 \pm \sqrt{36 + 48}}{2}

 x = \frac{-6 \pm \sqrt{84}}{2}

To calculate the decimal answers, work out each answer in turn on a calculator.

x = \frac{-6 \pm \sqrt{84}}{2}

The first solution is x = \frac{-6 + \sqrt{84}}{2} = 1.58 (3 sf).

The second solution is x = \frac{-6 - \sqrt{84}}{2} = -7.58 (3 sf).

Question

Solve 2x^2 - 10x + 3 = 0 using the quadratic formula.

Identify a, b and c. a = 2, b = -10 and c = 3.

Substitute these values into the quadratic equation x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

This gives:

x = \frac{--10 \pm \sqrt{(-10)^2 - 4 \times 2 \times 3}}{2 \times 2} (-10)^2 = -10 \times -10 = 100

x = \frac{10 \pm \sqrt{100 - 24}}{4}. --10 = 10 as two negatives make a positive.

x = \frac{10 \pm \sqrt{76}}{4}

x = \frac{10 + \sqrt{76}}{4} or x = \frac{10 - \sqrt{76}}{4}

x = 4.68 (3 sf) or x = 0.32 (3 sf)