Quadratic Equations

If the roots of the equation${\mathrm{x}}^2+2(3\mathrm{a}+5)\mathrm{x}+2(9{\mathrm{a}}^2+25)=0$ are rational, then the value of $\mathrm{a}$ is _____.

If x is real, the minimum value of 4x ² + 4 x+ 9 is

$\left(1+m^2\right)x^2+2 mcx+c^2-a^2=0$ has equal roots if ________.

What is the solution of the equation $\sqrt{\frac{x}{x+3}}-\sqrt{\frac{x+3}{x}}=-\frac{3}{2}$?

Quadratic equation corresponding to the roots $2+\sqrt{5} \& 2-\sqrt{5}$ is ______.

If $\left(–4\right)$ is one root of the equation $x^2+px-4=0$ and the equation $x^2+px+q=0$ has equal roots then the value of $p$ and $q$ are respectively.

Find the maximum value of $\frac{2x^2+3x+4}{x^2+x+3}?$

Which of the following equations has real roots$?$

What quantity should be added to both sides of this equation to complete the square?

$x^2-8x=5$

One factor of $5+8\mathrm{x}–4{\mathrm{x}}^2$ is $(2\mathrm{x}+1)$ then another factor is _____.

If $I\pm I^2$² are the roots of $x^2+px+12=0$ and $I\pm -I^2=1$ then its roots-

The roots of equation $y^3+y^2-y-1=0$ are

Toni is solving this equation by completing the square. $a{x}^2+bx+c=0$ (where $a\ge 0$) Step $1$: $a{x}^2+bx=-c$, Step $2$: $x^2+\frac{b}{a}x=-\frac{c}{a}$ and Step $3$:? Which should be the Step $3$ in the solution?

If the roots of $a{x}^2+bx+c=0$ are in the ratio $\frac{p}{q}$ then the value of $\frac{b^2}{\left(ca\right)}$ is______.

If $\alpha ,\beta$ are the roots of equation $x^2-5x+6=0$ and $\alpha >\beta$ then the equation with roots $(\alpha \beta +\alpha +\beta )$ and $(\alpha \beta -\alpha -\beta )$ is

If $\alpha ,\beta$ are the roots of equation $x^2-5x+6=0$ and $\alpha >\beta$ then the equation with roots $\left({\alpha }^2+\beta \right)$ and $\left(\alpha +{\beta }^2\right)$ is

Solving equation $\sqrt{x^2-9x+18}+\sqrt{x^2+2x-15}=\sqrt{x^2-4x+3}$ following roots are obtained

If the roots of the equation $2x^2+8x-m^3=0$ are equal then value of $m$ is _____.