Algebra · Junior Cert

Solving Quadratics

Junior Cert Higher · Factorising, the $-b$ formula & roots · Tap NEXT to begin

Section 1 of 4

Solve by Factorising

Get everything to one side $=0$, factorise, then set each bracket to $0$. The two answers are the roots.

Worked example

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

$(x+4)(x-2) = 0$

$x+4 = 0$ or $x-2 = 0$

$x = -4$ or $x = 2$

Worked example

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

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

$x = 5$ or $x = -2$

Worked example — the roots

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

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

$x = 2$ or $x = 3$

Worked example — rearrange first

Solve $x^2-4 = 3x$.

$x^2-3x-4 = 0 \Rightarrow (x-4)(x+1) = 0$

$x = 4$ or $x = -1$

Worked example

Solve $2x^2-x-10 = 0$.

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

$x+2 = 0$ or $2x-5 = 0$

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

Worked example — a fraction and a rearrange

Solve $\tfrac{1}{2}x^2 = x+4$.

$x^2 = 2x+8 \Rightarrow x^2-2x-8 = 0$

$(x-4)(x+2) = 0$

$x = 4$ or $x = -2$

You try

Solve $3x^2+16x = 12$.

Pen and paper out — try it before you reveal.

$3x^2+16x-12 = 0$; guide number $-36$, numbers $+18,-2$.

$3x(x+6)-2(x+6) = 0 \Rightarrow (x+6)(3x-2) = 0$

$x = -6$ or $x = \dfrac{2}{3}$

You try

Solve $x^2-9 = 0$ (a difference of squares).

Pen and paper out — try it before you reveal.

$x^2-3^2 = 0 \Rightarrow (x-3)(x+3) = 0$

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

Section 2 of 4

The −b Formula

When it will not factorise neatly, use the formula for $ax^2+bx+c = 0$:

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

Worked example

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

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

$x = \dfrac{2 \pm \sqrt{(-2)^2-4(1)(-5)}}{2} = \dfrac{2 \pm \sqrt{24}}{2}$

$\sqrt{24} = 2\sqrt{6}$, so $x = 1 \pm \sqrt{6}$

$x \approx 3.4$ or $x \approx -1.4$

Worked example — to 2 decimal places

Solve $x^2-3x-7 = 0$.

$a=1,\ b=-3,\ c=-7$

$x = \dfrac{3 \pm \sqrt{9+28}}{2} = \dfrac{3 \pm \sqrt{37}}{2}$

$x \approx 4.54$ or $x \approx -1.54$

You try

Solve $2x^2-3x-7 = 0$ to one decimal place.

Pen and paper out — try it before you reveal.

$a=2,\ b=-3,\ c=-7$

$x = \dfrac{3 \pm \sqrt{9+56}}{4} = \dfrac{3 \pm \sqrt{65}}{4}$

$x \approx 2.8$ or $x \approx -1.3$

You try

Solve $5x^2-x-9 = 0$ to one decimal place.

Pen and paper out — try it before you reveal.

$a=5,\ b=-1,\ c=-9$

$x = \dfrac{1 \pm \sqrt{1+180}}{10} = \dfrac{1 \pm \sqrt{181}}{10}$

$x \approx 1.4$ or $x \approx -1.2$

Section 3 of 4

Build a Quadratic From Its Roots

Reverse the process: if the roots are $p$ and $q$, the equation is $(x-p)(x-q) = 0$.

Worked example

Form a quadratic with roots $5$ and $6$.

$(x-5)(x-6) = 0$

$x^2-6x-5x+30 = 0$

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

Worked example

Form a quadratic with roots $-5$ and $8$.

$(x+5)(x-8) = 0$

$x^2-8x+5x-40 = 0$

$x^2-3x-40 = 0$

You try

Form a quadratic with roots $\tfrac{1}{2}$ and $\tfrac{2}{3}$.

Pen and paper out — try it before you reveal.

$x=\tfrac12 \Rightarrow 2x-1=0$; $x=\tfrac23 \Rightarrow 3x-2=0$

$(2x-1)(3x-2) = 0$

Section 4 of 4

Surds You’ll Meet

The formula often leaves a surd (a root that isn’t exact). Handy rules: $\sqrt{a}\,\sqrt{b} = \sqrt{ab}$ and $(\sqrt{a})^2 = a$.

$\sqrt{2}\times\sqrt{3} = \sqrt{6}$

$\sqrt{3}\times\sqrt{11} = \sqrt{33}$

$(2\sqrt{3})^2 = 4\times 3 = 12$

Worked example

Simplify $(3\sqrt{5})(2\sqrt{7})$.

Multiply the numbers, multiply the roots: $6\sqrt{35}$

$6\sqrt{35}$

Worked example

Simplify $(2\sqrt{3})(7\sqrt{3})$.

$14\sqrt{9} = 14\times 3$

$42$

You try

Simplify $(5\sqrt{2})(4\sqrt{7})$.

Pen and paper out — try it before you reveal.

$20\sqrt{2\times 7}$

$20\sqrt{14}$

Powers of a fraction split top and bottom: $\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$.

Worked example

Simplify $\left(\dfrac{2x}{3}\right)^3$.

$\dfrac{(2x)^3}{3^3}$

$\dfrac{8x^3}{27}$

You try

Simplify $\left(\dfrac{-3x}{2}\right)^2$.

Pen and paper out — try it before you reveal.

$\dfrac{(-3x)^2}{2^2}$

$\dfrac{9x^2}{4}$

That’s solving quadratics.

Factorise to $0$, or use the $-b$ formula, and you can even build a quadratic backwards from its roots.