Solving by quadratic formula - Higher
Using the quadratic formula is another method of solving quadratic equations. You will need to learn this formula, as well as understand how to use it.
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:
\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
\(\frac{(- 6) \pm \sqrt{6^2 - 4 \times 1 \times -12}}{2 \times 1}\)
\(\frac{(-6) \pm \sqrt{36 + 48}}{2}\)
\(\frac{(-6) \pm \sqrt{84}}{2}\)
To calculate the decimal answers, work out each answer in turn on a calculator.
\(\frac{(-6) \pm \sqrt{84}}{2}\)
The first solution is \(\frac{(-6) + \sqrt{84}}{2} = 1.58\) (2 dp).
The second solution is \(\frac{(-6) - \sqrt{84}}{2} = -7.58\) (2 dp).
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:
\(x = \frac{--10 \pm \sqrt{(-10)^2 - 4 \times 2 \times 3}}{2 \times 2}\)
\(\qquad (-10)^2 = -10 \times -10 = 100\)
\(x = \frac{10 \pm \sqrt{100 - 24}}{4}\)
\(\qquad b = -10\), therefore \(-b = --10 = 10\)\( \qquad \small \text{as two negatives make a positive}\)
\(x = \frac{10 \pm \sqrt{76}}{4}\)
\(x = \frac{10 + \sqrt{76}}{4}\) or \(x = \frac{10 - \sqrt{76}}{4}\)
\(x = 4.68\) (2 dp) or \(x = 0.32\) (2 dp)
Example
\(x^2 + 2x + 5 = 0\)
Using the quadratic formula to solve this equation with \(a = 1\), \(b = 2\) and \(c = 5\) gives:
\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
\(x = \frac{-2 \pm \sqrt{2^2 - 4 \times 1 \times 5}}{2 \times 1} = \frac{-2\pm \sqrt{-16}}{2}\)
It is not possible to find the square root of a negative number, so the equation has no real solutions.
The graph of \(y = x^2 + 2x + 5\) does not cross or touch the x-axis as the equation \(x^2 + 2x + 5 = 0\) has no roots.
More guides on this topic
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- NEW: Factorising expressions
- NEW: Algebraic reasoning and proof
- NEW: Formulae
- NEW: Higher – Functions
- NEW: Solving one- and two-step linear equations
- NEW: Solving more complex linear equations
- NEW: Inequalities
- NEW: Higher – Graphs of inequalities
- NEW: Sequences
- NEW: Geometric and special sequences
- NEW: How to plot straight line graphs
- NEW: Working out equations of a line and calculating gradient
- NEW: Equations of parallel and perpendicular lines
- Algebraic expressions - OCR
- Algebraic formulae - OCR
- Algebraic fractions - OCR
- Solving linear equations - OCR
- Solving simultaneous equations - OCR
- Inequalities - OCR
- Sequences - OCR
- Straight line graphs - OCR
- Other graphs - OCR
- Transformation of curves - Higher - OCR
- Using and interpreting graphs - OCR
