A quadratic curve is given by $y=hx^2-2x-9h+6$, where $h$ is a constant. Show that the equation

Question

A quadratic curve is given by $y=hx^2-2x-9h+6$, where $h$ is a constant. Show that the equation $y=0$ has real roots for all values of $h$ except $h=0$[3] State the value of $h$ in the case where $y=0$ has two real and equal roots.[1] Given that the line $y=2x-h-12$ meets the curve $y=hx^2-2x-9h+6$, find the range of values of $h$.[5]

Accepted Answer

(a) $D=4(3h-1)^2\ge0$, so $y=0$ has real roots for all $h\ne0$. (b) $h=\frac{1}{3}$. (c) $h\le\frac{1}{4},\ h\ne0$ or $h\ge2$.

Quadratic Functions, Equations and Inequalities - Nature of Roots of Quadratic Equations

Quadratic Functions, Equations and Inequalities - Nature of Roots of Quadratic Equations

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