The quadratic formula $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$for $ax^2+bx+c=0$; use it when factorising is hard or the roots are surds/decimals
Setup $ax^2+bx+c=0$rearrange to $=0$ FIRST, then read off $a$, $b$, $c$ WITH their signs
Discriminant $\Delta=b^2-4ac$the part under the root; its SIGN tells you the number of real roots
Nature of roots $b^2-4ac \;\begin{cases}>0\\=0\\<0\end{cases}$$>0$: two real roots · $=0$: one repeated root · $<0$: no real roots
Surd form · $x^2-7x-5=0$
$a=1,\ b=-7,\ c=-5$
$x=\dfrac{7\pm\sqrt{49+20}}{2}$
$=\dfrac{7\pm\sqrt{69}}{2}$
To 2 d.p. · $2x^2+8x+1=0$
$x=\dfrac{-8\pm\sqrt{64-8}}{4}=\dfrac{-8\pm\sqrt{56}}{4}$
$x=-0.13$ or $x=-3.87$
Nature · $x^2+3x+5=0$
$b^2-4ac=9-20=-11$
$-11<0$, so there are no real roots
Quadratic formula: the formula that solves any $ax^2+bx+c=0$.
Discriminant: $b^2-4ac$ – tells you how many real roots the equation has.
Repeated root: when $b^2-4ac=0$: the two roots are equal (one solution).
Coefficient: the numbers $a$, $b$, $c$ in $ax^2+bx+c$.
Surd form: an EXACT answer left with a root, e.g. $\dfrac{7\pm\sqrt{69}}{2}$.
✗ Using $+b$ instead of $-b$
✓ the formula starts with $-b$; for $b=-5$, $-b=+5$.
✗ Dividing only the root (or only $-b$) by $2a$
✓ the WHOLE numerator $-b\pm\sqrt{\;}$ is over $2a$.
✗ Forgetting signs in $b^2-4ac$ when $a$ or $c$ is negative
✓ $-4ac$ with $c=-5$ gives $+20$.
✗ Rounding too early
✓ keep the surd until the last step, then round.
• Everything is divided by $2a$ – not just $2$, and not just the root.
• It is $-b$ at the start: if $b=-7$ then $-b=+7$.
• The $\pm$ gives the TWO roots – write both.
• Write $a=\,,\ b=\,,\ c=\,$ before you substitute – it stops sign slips.
• Check the discriminant first: a negative means STOP, there are no real roots.
• If $b^2-4ac$ is a perfect square, the quadratic actually factorises.
mathedup.co.uk · sheet 7BHE