Question

Find the zeroes of the following quadratic polynomial and verify.
$2a^2-2\sqrt{2}a+1$

Answer 1

Zeroes of $2a^2-2\sqrt{2}a+1$

we can rewrite the quadratic polynomial as :

${\left(\sqrt{2}a\right)}^2-2\left(\sqrt{2}a\right)\left(1\right)+{\left(1\right)}^2$

this take the form of $a^2-2ab+b^2$

Therefore, ${(\sqrt{2}a-1)}^2=2a^2-2\sqrt{2}a+1$

Hence, the zeroes of ${(\sqrt{2}a-1)}^2$ can be found by equating ${(\sqrt{2}a-1)}^2=0$

$\Rightarrow \sqrt{2}a-1=0$

$\Rightarrow a=\frac{1}{\sqrt{2}}$

we can verify $a=\frac{1}{\sqrt{2}}$ as zero by substituting this value back to the original expression.

$\Rightarrow 2{\left(\frac{1}{\sqrt{2}}\right)}^2-2\sqrt{2}\left(\frac{1}{\sqrt{2}}\right)+1=2\left(\frac{1}{2}\right)-2\left(1\right)+1=0$

Hence, $\frac{1}{\sqrt{2}}$ is a zero of polynomial $2a^2-2\sqrt{2}a+1.$

Hence, the answer is $\frac{1}{\sqrt{2}}.$