Question

Find the quadratic polynomials in each case, with the given numbers as the sum and product of its zeroes respectively.
(i) $\frac{1}{4},-1$
(ii) $\sqrt{2},\frac{1}{4}$
(iii) $0,\sqrt{5}$
(iv) $1,1$
(v) $-\frac{1}{4},\frac{1}{4}$
(vi) $4,1$

Answer 1

General formula:

$P\left(x\right)=x^2-\left(\text{sum-of-roots}\right)x+\left(\text{product-of-roots}\right)$

(i)

$x^2-\left(\frac{1}{4}\right)x-1=0$

$\Rightarrow 4x^2-x-4=0$

(ii)

$x^2-\left(\sqrt{2}\right)x+\frac{1}{4}=0$

$\Rightarrow 4x^2-4\sqrt{2}x+1=0$

(iii)

$x^2-\left(0\right)x+\sqrt{5}=0$

$\Rightarrow x^2+\sqrt{5}=0$

(iv)

$x^2-\left(1\right)x+1=0$

$\Rightarrow x^2-x+1=0$

(v)

$x^2-(-\frac{1}{4})x+\frac{1}{4}=0$

$4x^2+x+1=0$

(vi)

$x^2-\left(4\right)x+1=0$

$\Rightarrow x^2-4x+1=0$