Question

level 4: open ended questions 8. complete the table

discriminant	number and type of root(s)	example sketch
$b^{2}-4ac = 0$		graph
$b^{2}-4ac < 0$		graph

Explanation:

Step1: Recall discriminant - root relationship

For a quadratic equation $ax^{2}+bx + c=0$ ($a
eq0$), the discriminant $\Delta=b^{2}-4ac$ determines the nature of the roots.

Step2: When $\Delta = 0$

If $b^{2}-4ac = 0$, the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ gives $x=\frac{-b\pm0}{2a}=-\frac{b}{2a}$. So, there is one real - root (a repeated root).

Step3: When $\Delta<0$

If $b^{2}-4ac<0$, the expression $\sqrt{b^{2}-4ac}$ is not a real number in the set of real numbers. In the complex - number system, the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ gives two complex conjugate roots of the form $x = p\pm qi$, where $p,q\in\mathbb{R}$ and $q
eq0$.

Answer:

Discriminant	Number and type of root(s)	Example sketch
$b^{2}-4ac = 0$	One real root (repeated)	[Given sketch]
$b^{2}-4ac<0$	Two complex conjugate roots	[Given sketch]