Substitution $y=mx+c \;\Rightarrow\; \text{replace } y$put the linear equation into the non-linear one to get one unknown
Form a quadratic $ax^2+bx+c=0$collect every term on one side $=0$, then factorise (or use the formula)
Line & circle $x^2+y^2=r^2$substitute $y=mx+c$ and expand $(mx+c)^2$ in full, then simplify
Discriminant test $b^2-4ac$$>0$: two points; $=0$: tangent (one point); $<0$: they never meet Line & quadratic · $y=x^2$, $\;y=x+2$
$x^2=x+2\Rightarrow x^2-x-2=0$
$(x-2)(x+1)=0$, so $x=2$ or $x=-1$
points $(2,4)$ and $(-1,1)$
Line & circle · $x^2+y^2=20$, $\;y=2x$
$x^2+(2x)^2=20\Rightarrow 5x^2=20$
$x^2=4$, so $x=\pm 2$
points $(2,4)$ and $(-2,-4)$
Tangent (discriminant) · $y=x+k$ touches $y=x^2$
$x^2-x-k=0$; one root, so $b^2-4ac=0$
$1+4k=0\Rightarrow k=-\tfrac{1}{4}$
Simultaneous: two equations that must both be true at the same point.
Intersection: a point where the line and curve cross – a solution pair $(x,y)$.
Tangent: a line that touches a curve at exactly one point (a repeated root).
Discriminant: $b^2-4ac$ – tells you how many real solutions the quadratic has.
Substitution: replacing $y$ (or $x$) using the linear equation, to get one equation in one unknown.
✗ Solving for $x$ and stopping there
✓ each $x$ needs its matching $y$ – put it back into the linear equation.
✗ Dividing $x^2-2x=0$ by $x$
✓ that loses the root $x=0$; factorise as $x(x-2)=0$ instead.
✗ Writing $(2x+1)^2=4x^2+1$
✓ expand the whole bracket: $(2x+1)^2=4x^2+4x+1$.
✗ Pairing the wrong $x$ with the wrong $y$
✓ keep each pair together – substitute one $x$ at a time.
• A line can cut a curve at $2$ points, touch it at $1$ (a tangent), or miss it entirely – the discriminant tells you which.
• Always find both coordinates of each point: an $x$-value on its own is only half the answer.
• Substitute each point back into both original equations to check.
• Rearrange the linear equation for $y$ (or $x$) first, then substitute – it is the tidiest route.
• Get the quadratic to "$=0$" before you factorise.
• Two solutions is the usual answer; one repeated solution means the line is a tangent.
mathedup.co.uk · sheet W8DR