Question

Find the zeroes of the quadratic polynomial $x^2-5$ and verify the relationship between the zeroes and the coefficients.

Hint:

For $x^2+bx+c$, the zeroes' sum is $-b$ and product is $c$. Spot $b$ and $c$ quickly to verify without fully solving.

Answer 1

Step 1: Find the zeroes.
Solve $x^2-5=0 $\Rightarrow$ x=\pm\sqrt{5}$. Thus the zeroes are $\alpha=\sqrt{5}$ and $\beta=-\sqrt{5}$.

Step 2: Verify relationships.
For $ax^2+bx+c$, sum of zeroes $=\,-\dfrac{b}{a}$ and product $=\,\dfrac{c}{a}$.
Here $a=1,\ b=0,\ c=-5$.
Sum: $\alpha+\beta=\sqrt{5}+(-\sqrt{5})=0\ = -\dfrac{b}{a}=-\dfrac{0}{1}=0$.
Product: $\alpha\beta=\sqrt{5}\cdot(-\sqrt{5})=-5\ = \dfrac{c}{a}=\dfrac{-5}{1}=-5$. \boxed{\text{Zeroes: }\sqrt{5},\ -\sqrt{5}\ \ \text{and both relations are verified.}}