VT · The Quadratic Formula

In each set, one thing changes and everything else stays the same. Work them out in order and look for the pattern — the last line tells you what to notice.

①Only $c$ changes – watch the discriminantchanging: the constant $c$

Work out the discriminant $b^2-4ac$ and state the number of real roots.

$x^2+4x+2=0$

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

$x^2+4x+4=0$

$x^2+4x+5=0$

As $c$ grows, the discriminant falls: two roots, then one, then none. Where is the tipping point?

②Only $b$ changes – solve in surd formchanging: the coefficient $b$

Solve using the formula, giving each answer in exact (surd) form.

$x^2+3x+1=0$

$x^2+4x+1=0$

$x^2+5x+1=0$

One of these has a surd that simplifies to remove the fraction. Spot which, and why.

③Equal roots – find $k$changing: the constant term

Each equation has equal (repeated) roots. Work out the possible values of $k$.

$x^2+kx+4=0$

$x^2+kx+9=0$

$x^2+kx+16=0$

$x^2+kx+25=0$

Equal roots means the discriminant is $0$, so $k^2=4c$. How does $k$ depend on $c$?

Answers · The Quadratic Formula

Variation practice

① Only $c$ changes – watch the discriminant

$x^2+4x+2=0$: $8,\ \text{two roots}$$x^2+4x+3=0$: $4,\ \text{two roots}$$x^2+4x+4=0$: $0,\ \text{one repeated root}$$x^2+4x+5=0$: $-4,\ \text{no real roots}$

② Only $b$ changes – solve in surd form

$x^2+3x+1=0$: $\dfrac{-3\pm\sqrt5}{2}$$x^2+4x+1=0$: $-2\pm\sqrt3$$x^2+5x+1=0$: $\dfrac{-5\pm\sqrt{21}}{2}$

③ Equal roots – find $k$

$x^2+kx+4=0$: $k=\pm4$$x^2+kx+9=0$: $k=\pm6$$x^2+kx+16=0$: $k=\pm8$$x^2+kx+25=0$: $k=\pm10$