Quadratic Equations — Mathslive.ie

Section 1 of 2

Solving by Factorising

A quadratic equation has an $x^2$ in it, so it can have two answers. The key fact: if two things multiplied together give zero, then at least one of them must be zero.

The zero rule

If $A \times B = 0$, then $A = 0$ or $B = 0$. So factorise, set the whole thing $= 0$, and put each bracket to zero.

Worked example — a trinomial

Solve $x^2 + 7x + 12 = 0$.

$(x + 3)(x + 4) = 0$

$x + 3 = 0$ or $x + 4 = 0$

$x = -3$ or $x = -4$

Worked example — another

Solve $x^2 - 8x + 15 = 0$.

Two numbers multiply to $15$, add to $-8$: $-3$ and $-5$

$(x - 3)(x - 5) = 0$

$x = 3$ or $x = 5$

Worked example — no constant term

Solve $x^2 - 6x = 0$.

Take out the common factor $x$: $x(x - 6) = 0$

$x = 0$ or $x - 6 = 0$

$x = 0$ or $x = 6$

You try

Solve $2x^2 - 3x = 0$.

No constant, so take out the common factor $x$. One root is automatically $0$.

$x(2x - 3) = 0$

$x = 0$ or $2x - 3 = 0$

$x = 0$ or $x = \tfrac{3}{2}$

Section 2 of 2

Difference of Squares & the Formula

Difference of two squares

$x^2 - y^2 = (x + y)(x - y)$ — spot it when there is a square minus a square (no middle term).

Worked example — a difference of squares

Solve $x^2 - 9 = 0$.

$x^2 - 3^2 = 0$

$(x - 3)(x + 3) = 0$

$x = 3$ or $x = -3$

You try

Solve $4x^2 - 25 = 0$.

Both are perfect squares: $4x^2 = (2x)^2$ and $25 = 5^2$.

$(2x)^2 - 5^2 = 0$

$(2x - 5)(2x + 5) = 0$

$x = \tfrac{5}{2}$ or $x = -\tfrac{5}{2}$

The −b formula

When it won’t factorise into neat brackets, use $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Read off $a, b, c$ and substitute — brackets around the negatives.

Worked example — use the formula

Solve $x^2 - 2x - 24 = 0$.

$a = 1,\ b = -2,\ c = -24$

$x = \dfrac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-24)}}{2(1)} = \dfrac{2 \pm \sqrt{4 + 96}}{2}$

$= \dfrac{2 \pm \sqrt{100}}{2} = \dfrac{2 \pm 10}{2}$

$x = 6$ or $x = -4$

That’s Algebra — all eight classes.

Solve quadratics by factorising, by difference of squares, and by the $-b$ formula. You’ve covered the whole Ordinary Level algebra course.