Quadratic Equation and Its Solutions

Let $P\left(x\right)=x^2+bx+c$, where $b$ and $c$ are integers. If $P\left(x\right)$ is a factor of both $x^4+6x^2+25$ and $3x^4+4x^2+28x+5$, then the value of $P\left(1\right)$ is

If $x^2-x \cos \left(A+B\right)+1$ is a factor of the expression, $2x^4+4x^3\sin A \sin B-x^2\left(\cos 2A+\cos 2B\right)+4 x \cos A \cos B-2$. Then find the other factor.

Express the given Polynomial as a Product Of Linear Factors:

$f\left(x\right)=x^3+x^2-10x+8$

Identify the common factors in the given polynomials.

$x^2+7x+12,$ and $x^2+6x+8$

Form a quadratic equation, whose root's sum and product are $-3$ and $2$, respectively.

Form a quadratic equation, whose root's sum and product are $-3$ and $2$, respectively.

Express the given Polynomial as a Product Of Linear Factors:

$f\left(x\right)=x^4-x^3-8x^2+12x?$

Solve the equation $x^2-4x+13=0$ by factorisation method.

If $\mathrm{\alpha },\mathrm{\beta }$ and $\mathrm{\gamma },\mathrm{\delta }$ are the roots of $x^{\mathit{2}}\mathit{+}px\mathit{-}r\mathit{=}\mathit{0}$ and $x^2+px+r=0$ respectively, prove that $\left(\mathrm{\alpha }-\mathrm{\gamma }\right)\left(\mathrm{\alpha }-\mathrm{\delta }\right)=\left(\mathrm{\beta }-\mathrm{\gamma }\right)\left(\mathrm{\beta }-\mathrm{\delta }\right)$.

If $\mathrm{\alpha },\mathrm{\beta }$ and $\gamma ,\delta$ are the roots $x^2-ax+b=0$ and $x^2-bx+a=0$, find the value of $\left(\mathrm{\alpha }-\mathrm{\gamma }\right)\left(\mathrm{\beta }-\mathrm{\delta }\right)+\left(\mathrm{\alpha }-\mathrm{\delta }\right)\left(\mathrm{\beta }-\mathrm{\gamma }\right)$ 

If the roots of the equations $x^2-ax+b=0$ and $x^2-px+q=0$ be $\mathrm{\alpha },\mathrm{\beta }$ and $\gamma ,\delta$ respectively, form the quadratic equation whose roots are $\left(\mathrm{\alpha \gamma }+\mathrm{\beta \delta }\right)$ and $\left(\mathrm{\alpha \delta }+\mathrm{\beta \gamma }\right)$.

The Coefficient of $x$ in the quadratic equation $x^2+px+q=0$ was taken as $17$ in place of $13$ and its roots were found its roots were found to be $-2$ and $-15$. Find the roots of the original equation.

If $\mathrm{\alpha }$  and $\mathrm{\beta }$ are the roots of the equation $a{x}^2+2bx+c=0$ and $\left(\alpha +\delta \right)$ and $\left(\mathrm{\beta }+\mathrm{\delta }\right)$ are the roots of the equation $A{x}^2+2Bx+C=0$, then prove that $\frac{b^2-ac}{B^2-AC}=\frac{a^2}{A^2}$.

If the difference of the roots of the equation $x^2-px+q=0$ be the same as that of the equation $x^2-qx+p=0$, show that $p+q+4=0\left(p\ne q\right)$.

If there is a common root of $x^2+ax+b=0$ and $x^2+bx+a=0\left(a\ne b\right)$ have a common root, show that the other roots are the roots of the equation $x^2+x+ab=0$

For two unequal real numbers $p$ and $q$ if $x^2+px+q=0$ and $x^2+qx+p=0$ have a common root, show that $p+q+1=0$. Also show that the other roots of the above equations are the roots of the equation $x^2+x+pq=0$.

Show that $1$ is a root of $a\left(b-c\right)x^2+b\left(c-a\right)x+c\left(a-b\right)=0$; .

If one root of the quadratic equation $a{x}^2+bx+c=0$ be square of the other, then show that $b^3+a{c}^2+a^{2}c=3abc$.

In the equation $a{x}^1+bx+c=0$, one root is the square of the other, prove that $c{(a-b)}^3=a{(c- b)}^3$.

If one root of the equation $b{x}^2+cx+a=0$ be the square of the other, prove that $c^3+ab(a+b)=3abc$.