Singapore-Cambridge A-Level Mathematics (H2)

## Completing the square Any quadratic $ax^2 + bx + c$ with $a \neq 0$ can be written as $a(x + p)^2 + q$. Taking out $a$ from the first two terms and halving the coefficient of $x$ gives $$ ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 + c - \frac{b^2}{4a}. $$ The square is never negative, so if $a > 0$ the least value of the expression is $c - \frac{b^2}{4a}$, reached when $x = -\frac{b}{2a}$. If $a < 0$ the same value is the greatest value. The turning point of the graph is therefore $\left(-\frac{b}{2a},\; c - \frac{b^2}{4a}\right)$ and the line $x = -\frac{b}{2a}$ is its axis of symmetry. For $f(x) = 2x^2 - 12x + 7$: $$ \begin{aligned} f(x) &= 2(x^2 - 6x) + 7 \\ &= 2\left[(x - 3)^2 - 9\right] + 7 \\ &= 2(x - 3)^2 - 11. \end{aligned} $$ The least value is $-11$, at $x = 3$, and the graph is a $\cup$-shaped curve with vertex $(3, -11)$. Setting the completed form equal to zero and solving gives the quadratic formula, $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. $$ ## The discriminant The expression under the square root, $b^2 - 4ac$, is the **discriminant**, written $D$. It decides how many real roots $ax^2 + bx + c = 0$ has. | Discriminant | Roots | Graph of $y = ax^2 + bx + c$ | |---|---|---| | $D > 0$ | two distinct real roots | cuts the $x$-axis twice | | $D = 0$ | two equal real roots | touches the $x$-axis | | $D < 0$ | no real roots | does not meet the $x$-axis | If $a > 0$ and $D < 0$, the graph lies wholly above the $x$-axis, so $ax^2 + bx + c > 0$ for every…