The quadratic is the most examined object in Paper 1: completing the square gives the vertex and the range, the discriminant decides how many roots (and whether a line is a tangent), and the relations between roots and coefficients give quick routes to 'find \(\alpha^2 + \beta^2\)' questions.
Contents: Three forms of a quadratic (MM1.3, M4.11-M4.13)The discriminant $b^2 - 4ac$Roots and coefficientsMust-knowTrapsReal questions
Three forms of a quadratic (MM1.3, M4.11-M4.13)
- Standard \(ax^2 + bx + c\): \(y\)-intercept \(c\); the graph opens up if \(a > 0\) and down if \(a < 0\).
- Factorised \(a(x - p)(x - q)\): roots \(p\), \(q\); the axis of symmetry is \(x = \frac{p + q}{2}\).
- Completed square \(a(x - h)^2 + k\): vertex \((h, k)\), which is the minimum (if \(a > 0\)) or maximum (if \(a < 0\)). For \(x^2 + bx + c\): \(h = -\frac b2\), \(k = c - \frac{b^2}{4}\).
- The quadratic formula \(x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) must be recalled (no formula booklet).
The discriminant $b^2 - 4ac$
- \(> 0\): two distinct real roots; \(= 0\): one repeated root (the graph touches the axis); \(< 0\): no real roots.
- 'The line meets the curve at exactly one point' means: substitute, form a quadratic in \(x\), set the discriminant to zero.
- '\(f(x) > 0\) for all \(x\)' means \(a > 0\) and discriminant \(< 0\).
- Conditions on roots: both roots positive needs discriminant \(\geq 0\), sum \(-\frac ba > 0\) and product \(\frac ca > 0\).
Roots and coefficients
- If the roots are \(\alpha, \beta\): \(\alpha + \beta = -\dfrac ba\) and \(\alpha\beta = \dfrac ca\).
- \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta\) and \((\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta = \dfrac{b^2 - 4ac}{a^2}\).
- Disguised quadratics: \(x^4 - 5x^2 + 4\), \((x^2 - 3x)^2 - 2(x^2 - 3x) - 8\), \(9^x - 4\cdot 3^x + 3\) - substitute and remember to translate back (and to count the solutions correctly).
Must-know (no formula booklet)
- \(x^2 + bx + c = (x + \frac b2)^2 + c - \frac{b^2}{4}\)
- \(b^2 - 4ac\): \(>0\), \(=0\), \(<0\)
- \(\alpha + \beta = -b/a\), \(\alpha\beta = c/a\)
- vertex of \(a(x-h)^2 + k\) is \((h, k)\)
Classic traps
- When \(a\) is negative, completing the square gives a maximum, not a minimum.
- 'No real roots' for \(kx^2 + 4x + k = 0\): check whether \(k = 0\) makes the equation linear.
- Counting solutions of \(x^4 - 5x^2 + 4 = 0\): each positive value of \(x^2\) gives two values of \(x\).
Worked example
Q. For which \(k\) does \(x^2 + (k - 2)x + (k + 1) = 0\) have distinct real roots?
A. \((k-2)^2 - 4(k+1) = k^2 - 8k = k(k - 8) > 0\), so \(k < 0\) or \(k > 8\).