Quadratic Equations — Mathslive.ie

Section 1 of 4

Solve by Factorising (Guide Number)

The zero rule

If $(\ )(\ ) = 0$ then one bracket must be $0$. So factorise with the Guide Number, then set each bracket to zero.

★ The Guide Number method

1. Guide Number GN = a × c (first number times the last, keep the sign).
2. List the factor pairs of the GN.
3. Pick the pair that gives b (the middle): Add if GN is +, Sub if GN is −.
4. Split the middle, then group and take out the common bracket.

Worked example

Solve $x^2 + 5x + 6 = 0$.

GN $= 1\times6 = 6$ (Add)

Factor pairs of $6$: $1\times6$, $2\times3$

The pair that adds to $5$: $3$ and $2$

Split: $x^2 + 3x + 2x + 6$

Group: $x(x + 3) + 2(x + 3)$

$(x + 2)(x + 3) = 0$, so each bracket $= 0$

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

Worked example

Solve $x^2 - 11x + 30 = 0$.

GN $= 1\times30 = 30$ (Add)

Factor pairs of $30$: $1\times30$, $2\times15$, $3\times10$, $5\times6$

The pair that adds to $-11$: $-6$ and $-5$

Split: $x^2 - 6x - 5x + 30$

Group: $x(x - 6) - 5(x - 6)$

$(x - 5)(x - 6) = 0$, so each bracket $= 0$

$x = 5$ or $x = 6$

Worked example

Solve $x^2 + 5x - 6 = 0$.

GN $= 1\times(-6) = -6$ (Sub)

Factor pairs of $6$: $1\times6$, $2\times3$

The pair that subtracts to $5$: $-1$ and $6$

Split: $x^2 - x + 6x - 6$

Group: $x(x - 1) + 6(x - 1)$

$(x + 6)(x - 1) = 0$, so each bracket $= 0$

$x = -6$ or $x = 1$

Worked example

Solve $3x^2 - 10x + 8 = 0$.

GN $= 3\times8 = 24$ (Add)

Factor pairs of $24$: $1\times24$, $2\times12$, $3\times8$, $4\times6$

The pair that adds to $-10$: $-6$ and $-4$

Split: $3x^2 - 6x - 4x + 8$

Group: $3x(x - 2) - 4(x - 2)$

$(3x - 4)(x - 2) = 0$, so each bracket $= 0$

$x = \tfrac{4}{3}$ or $x = 2$

Worked example

Solve $3x^2 - 2x - 8 = 0$.

GN $= 3\times(-8) = -24$ (Sub)

Factor pairs of $24$: $1\times24$, $2\times12$, $3\times8$, $4\times6$

The pair that subtracts to $-2$: $-6$ and $4$

Split: $3x^2 - 6x + 4x - 8$

Group: $3x(x - 2) + 4(x - 2)$

$(3x + 4)(x - 2) = 0$, so each bracket $= 0$

$x = -\tfrac{4}{3}$ or $x = 2$

You try

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

GN $= 1\times12 = 12$ (Add). Factor pairs: $1\times12$, $2\times6$, $3\times4$.

GN $= 1\times12 = 12$ (Add)

Factor pairs of $12$: $1\times12$, $2\times6$, $3\times4$

The pair that adds to $7$: $4$ and $3$

Split: $x^2 + 4x + 3x + 12$

Group: $x(x + 4) + 3(x + 4)$

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

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

You try

Solve $x^2 - 12x + 20 = 0$.

GN $= 1\times20 = 20$ (Add). Factor pairs: $1\times20$, $2\times10$, $4\times5$.

GN $= 1\times20 = 20$ (Add)

Factor pairs of $20$: $1\times20$, $2\times10$, $4\times5$

The pair that adds to $-12$: $-10$ and $-2$

Split: $x^2 - 10x - 2x + 20$

Group: $x(x - 10) - 2(x - 10)$

$(x - 2)(x - 10) = 0$

$x = 2$ or $x = 10$

You try

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

GN $= 2\times(-6) = -12$ (Sub). Factor pairs: $1\times12$, $2\times6$, $3\times4$.

GN $= 2\times(-6) = -12$ (Sub)

Factor pairs of $12$: $1\times12$, $2\times6$, $3\times4$

The pair that subtracts to $-1$: $-4$ and $3$

Split: $2x^2 - 4x + 3x - 6$

Group: $2x(x - 2) + 3(x - 2)$

$(2x + 3)(x - 2) = 0$

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

Section 2 of 4

No Constant — Common Factor

If there is no number on its own, just take out the common factor $x$ — no Guide Number needed.

Worked example

Solve $3x^2 + 5x = 0$.

No constant → take out the common factor $x$

$x(3x + 5) = 0$

$x = 0$ or $3x + 5 = 0$

$x = 0$ or $x = -\tfrac{5}{3}$

Worked example

Solve $5x^2 - 7x = 0$.

No constant → take out the common factor $x$

$x(5x - 7) = 0$

$x = 0$ or $5x - 7 = 0$

$x = 0$ or $x = \tfrac{7}{5}$

Worked example

Solve $3x^2 - 11x = 0$.

No constant → take out the common factor $x$

$x(3x - 11) = 0$

$x = 0$ or $3x - 11 = 0$

$x = 0$ or $x = \tfrac{11}{3}$

You try

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

No constant → take out the common factor (no GN needed).

Take out the common factor $x$

$x(2x + 3) = 0$

$x = 0$ or $2x + 3 = 0$

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

You try

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

No constant → take out the common factor (no GN needed).

Take out the common factor $x$

$x(x - 4) = 0$

$x = 0$ or $x - 4 = 0$

$x = 0$ or $x = 4$

Section 3 of 4

Difference of Two Squares

Worked example

Solve $x^2 - 49 = 0$.

No middle term and a minus → difference of two squares

Write as $(x)^2 - 7^2$

$(x - 7)(x + 7) = 0$

$x = 7$ or $x = -7$

Worked example

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

No middle term and a minus → difference of two squares

Write as $(2x)^2 - 5^2$

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

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

Worked example

Solve $9x^2 - 49 = 0$.

No middle term and a minus → difference of two squares

Write as $(3x)^2 - 7^2$

$(3x - 7)(3x + 7) = 0$

$x = \tfrac{7}{3}$ or $x = -\tfrac{7}{3}$

You try

Solve $25x^2 - 16 = 0$.

Factorise as $(5x-4)(5x+4)=0$.

No middle term → difference of two squares

$(5x - 4)(5x + 4) = 0$

Each bracket $=0$

$x = \tfrac{4}{5}$ or $x = -\tfrac{4}{5}$

You try

Solve $x^2 - 25 = 0$.

Factorise as $(x-5)(x+5)=0$.

No middle term → difference of two squares

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

Each bracket $=0$

$x = 5$ or $x = -5$

Section 4 of 4

The Quadratic Formula

When it won’t factorise (in your tables)

$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Use it when the Guide Number won’t give whole numbers.

Worked example

Solve $x^2 - 2x - 7 = 0$ to one decimal place.

$a=1,\ b=-2,\ c=-7$

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

$x = \dfrac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-7)}}{2(1)}$

$x = \dfrac{2 \pm \sqrt{32}}{2}$

$x = 3.8$ or $x = -1.8$

Worked example

Solve $x^2 - 4x + 1 = 0$ to one decimal place.

$a=1,\ b=-4,\ c=1$

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

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

$x = \dfrac{4 \pm \sqrt{12}}{2}$

$x = 3.7$ or $x = 0.3$

Worked example

Solve $x^2 + 3x - 3 = 0$ to one decimal place.

$a=1,\ b=3,\ c=-3$

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

$x = \dfrac{-(3) \pm \sqrt{(3)^2 - 4(1)(-3)}}{2(1)}$

$x = \dfrac{-3 \pm \sqrt{21}}{2}$

$x = 0.8$ or $x = -3.8$

You try

Solve $x^2 - 5x + 2 = 0$ to one decimal place.

Won’t factorise → use the formula. $a=1,\ b=-5,\ c=2$.

$a=1,\ b=-5,\ c=2$

$x = \dfrac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(2)}}{2(1)}$

$x = \dfrac{5 \pm \sqrt{17}}{2}$

$x = 4.6$ or $x = 0.4$

You try

Solve $2x^2 + 3x - 4 = 0$ to one decimal place.

Won’t factorise → use the formula. $a=2,\ b=3,\ c=-4$.

$a=2,\ b=3,\ c=-4$

$x = \dfrac{-(3) \pm \sqrt{(3)^2 - 4(2)(-4)}}{2(2)}$

$x = \dfrac{-3 \pm \sqrt{41}}{4}$

$x = 0.9$ or $x = -2.4$

You try

Solve $x^2 - 6x + 2 = 0$ to one decimal place.

Won’t factorise → use the formula. $a=1,\ b=-6,\ c=2$.

$a=1,\ b=-6,\ c=2$

$x = \dfrac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(2)}}{2(1)}$

$x = \dfrac{6 \pm \sqrt{28}}{2}$

$x = 5.6$ or $x = 0.4$

You try

Solve $3x^2 - 2x - 4 = 0$ to one decimal place.

Won’t factorise → use the formula. $a=3,\ b=-2,\ c=-4$.

$a=3,\ b=-2,\ c=-4$

$x = \dfrac{-(-2) \pm \sqrt{(-2)^2 - 4(3)(-4)}}{2(3)}$

$x = \dfrac{2 \pm \sqrt{52}}{6}$

$x = 1.5$ or $x = -0.9$

Forming a quadratic from its roots

Worked example — from roots

Write the quadratic with roots $2$ and $3$.

$(x-2)(x-3)=0$

$x^2 - 5x + 6 = 0$

You try

Write the quadratic with roots $-1$ and $4$.

$(x+1)(x-4)=0$, then expand.

$(x+1)(x-4)=0$

$x^2 - 3x - 4 = 0$

That’s Quadratic Equations.

The Guide Number for factorising, common factor when there’s no constant, difference of squares, and the formula.