Let \(f\left( x \right) = p{x^2} + qx - 4p\), where p ≠ 0. Find Find the number of roots for the equation \(f\left( x \right) = 0\).

Justify your answer.

METHOD 1

evidence of discriminant      (M1)
eg  \({b^2} - 4ac,\,\,\Delta \)

correct substitution into discriminant      (A1)
eg  \({q^2} - 4p\left( { - 4p} \right)\)

correct discriminant       A1
eg  \({q^2} + 16{p^2}\)

\(16{p^2} > 0\,\,\,\,\left( {{\text{accept}}\,\,{p^2} > 0} \right)\)     A1

\({q^2} \geqslant 0\,\,\,\,\left( {{\text{do not accept}}\,\,{q^2} > 0} \right)\)     A1

\({q^2} + 16{p^2} > 0\)      A1

\(f\) has 2 roots     A1 N0

METHOD 2

y-intercept = −4p (seen anywhere)      A1

if p is positive, then the y-intercept will be negative      A1

an upward-opening parabola with a negative y-intercept      R1
eg  sketch that must indicate p > 0.

if p is negative, then the y-intercept will be positive      A1

a downward-opening parabola with a positive y-intercept      R1
eg  sketch that must indicate p > 0.

\(f\) has 2 roots     A2 N0

[7 marks]