1 Quadratic Equations in One Unknown
1.1 The discriminant
A quadratic equation in one unknown has the form
\[ ax^2 + bx + c = 0, \qquad a \neq 0, \tag{1.1}\]
where \(a\), \(b\) and \(c\) are real. The quantity
\[ \Delta = b^2 - 4ac \]
is called the discriminant of Equation 1.1, and it determines the nature of the roots.
| Discriminant | Nature of roots |
|---|---|
| \(\Delta > 0\) | Two distinct real roots |
| \(\Delta = 0\) | One double (repeated) real root |
| \(\Delta < 0\) | No real roots (two complex roots) |
The roots of Equation 1.1 are
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \]
1.2 Worked example
Example 1.1 (Finding a range of values of \(k\)) The equation \(x^2 + kx + (k + 3) = 0\) has two distinct real roots. Find the range of values of \(k\).
Solution. Here \(a = 1\), \(b = k\) and \(c = k + 3\), so
\[ \Delta = k^2 - 4(1)(k + 3) = k^2 - 4k - 12 = (k - 6)(k + 2). \]
Two distinct real roots means \(\Delta > 0\), so \((k - 6)(k + 2) > 0\). The product is positive when both factors are positive or both are negative, giving
\[ k < -2 \quad \text{or} \quad k > 6. \]
1.3 Relations between roots and coefficients
If \(\alpha\) and \(\beta\) are the roots of Equation 1.1, then
\[ \alpha + \beta = -\frac{b}{a}, \qquad \alpha\beta = \frac{c}{a}. \]
These identities let you answer questions about the roots without ever solving the equation.
Example 1.2 (Using the sum and product of roots) If \(\alpha\) and \(\beta\) are the roots of \(2x^2 - 5x + 1 = 0\), find \(\alpha^2 + \beta^2\).
Solution. We have \(\alpha + \beta = \tfrac{5}{2}\) and \(\alpha\beta = \tfrac{1}{2}\). Then
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = \left(\frac{5}{2}\right)^2 - 2\left(\frac{1}{2}\right) = \frac{25}{4} - 1 = \frac{21}{4}. \]
1.4 Exercises
- Solve \(3x^2 - 7x + 2 = 0\) by factorisation.
- The equation \(x^2 - 2mx + (m + 6) = 0\) has a double root. Find all possible values of \(m\).
- If \(\alpha\) and \(\beta\) are the roots of \(x^2 + 4x - 9 = 0\), find \(\dfrac{1}{\alpha} + \dfrac{1}{\beta}\).
- Form a quadratic equation whose roots are \(3 + \sqrt{2}\) and \(3 - \sqrt{2}\).
Answers
- \(x = \tfrac{1}{3}\) or \(x = 2\).
- \(m = -2\) or \(m = 3\).
- \(\tfrac{4}{9}\).
- \(x^2 - 6x + 7 = 0\).