Q.If the equations $x^2-ax+b=0$ and $x^2-ex+f=0$ have one root in common and if the second equation has equal roots, then prove that $ae=2(b+f)$.

For $P(x)=ax^2+bx+c$ ($a\ne0$), completing the square gives the quadratic formula:

Discriminant $D=b^2-4ac$: $D>0$ gives two distinct real roots (parabola crosses the x-axis twice); $D=0$ gives one repeated real root (touches once); $D<0$ gives no real roots (a complex conjugate pair; parabola never meets the x-axis).

Sum and product of roots. For roots $\alpha,\beta$ of $ax^2+bx+c=0$: $\alpha+\beta=-\dfrac ba$, $\alpha\beta=\dfrac ca$. This pair of formulas is the single most useful tool in the chapter for:

• Building an equation from its roots: $x^2-(\alpha+\beta)x+\alpha\beta=0$;
• Transforming roots (reciprocals $1/\alpha,1/\beta$; doubles $2\alpha,2\beta$; negatives $-\alpha,-\beta$) into a new equation without ever solving for $\alpha,\beta$ explicitly; …