Solving quadratic equations

What a quadratic equation is

• A quadratic equation has $x^2$ as its highest power, written in standard form as $ax^2 + bx + c = 0$ (with $a \neq 0$).
• Its solutions are called roots — the $x$-values where the matching parabola crosses the $x$-axis.
• There are three methods; pick based on the equation:
  • Factorising — quickest, when the quadratic factorises neatly.
  • Quadratic formula — always works, especially when it does not factorise.
  • Completing the square — for exact answers and for finding a parabola's turning point.
• All three assume the equation equals zero, so rearrange to standard form first.

Solving by factorising

• Rearrange so the equation equals zero.
• Factorise into two brackets.
• Set each bracket to zero (the null factor law: if a product is 0, one factor must be 0) and solve.
• $x^2 - x - 6 = 0 \Rightarrow (x - 3)(x + 2) = 0 \Rightarrow x = 3$ or $x = -2$.

The quadratic formula

• For any $ax^2 + bx + c = 0$:

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

• $a$, $b$ and $c$ are the coefficients read straight from standard form.
• The $\pm$ ("plus or minus") produces the two roots — one using $+$, one using $-$.
• Use it when the quadratic does not factorise neatly.

Completing the square

• Completing the square rewrites $x^2 + bx + c$ as a perfect-square bracket plus a number:

$x^2 + bx + c = \left(x + \tfrac{b}{2}\right)^2 - \left(\tfrac{b}{2}\right)^2 + c$

• Method: halve the coefficient of $x$ to get the number inside the bracket, then subtract that number squared to keep the value the same.
• It is useful for solving (rearrange to get $x$ alone) and for reading off a parabola's turning point.

The discriminant

• The discriminant is the part under the root in the formula: $b^2 - 4ac$.
• It tells you how many real roots there are, without solving fully:
  • $b^2 - 4ac > 0$ → two real roots (the curve crosses the $x$-axis twice).
  • $b^2 - 4ac = 0$ → one repeated root (the curve just touches the $x$-axis).
  • $b^2 - 4ac < 0$ → no real roots (you cannot square-root a negative, so the curve misses the $x$-axis).