Embibe Experts Solutions for Exercise 1: Exercise

If $P\left(x\right)=a{x}^2+bx+c$ and $Q\left(x\right)=-a{x}^2+dx+c$ and $ac\ne 0$. Then prove that the product $P\left(x\right)Q\left(x\right)=0$ has at least two real roots.

If $p$ and $q$ are the roots of $4x^2-(5a+1)x+5a=0$ and $q=1+p$. Find the values of $a,p$ and $q$

If every pair of equations $x^2+px+qr=0$, $x^2+qx+rp=0$ and $x^2+rx+pq=0$ have a common root. Find sum and product of those common roots.

If $\alpha ,\beta$ are the roots of $x^2+px+1=0$ and $\gamma ,\delta$ are the roots of $x^2+qx+1=0,$ show that $(\alpha -\gamma )(\beta -\gamma )(\alpha +\delta )(\beta +\delta )=q^2-p^2$.

Find the condition for which both roots of quadratic equation $a{x}^2+bx+c=0$, $\left(a\ne 0\right)$ is zero.

If both roots of $a{x}^2+bx+c=0$ are negative, then what can you say about the signs of $a, b$ and $c$?

If $\alpha ,\beta$ are the roots of $x^2-px+q=0$, find the quadratic equation with roots $\frac{1}{p\alpha -q},\frac{1}{p\beta -q}$.

Find the greatest value of $\frac{x+2}{2x^2+3x+6}$ for real $x$.