Solution of a Quadratic Equation

If $\alpha$ and $\beta$ are the roots of the equation $x^2-4x+1=0$ the value of ${\alpha }^3+{\beta }^3$ is

The minimum value of the polynomial $p\left(x\right)=3x^2-5x+2$.

In a garden trees are planted in rows. The number of trees in each row is $5$ less than the number of rows. If the total number of trees planted be $336$. Find the number of trees planted in each row?

$I. 12x^2-8x-7=0$

$II. 10y^2+23y+12=0$

Write Shridhar Acharya's formula to find roots of quadratic equation $a{x}^2+bx+c=0$.

Divide $16$ into two parts, such that $2$ times the square of larger part is $164$ more than the square of smaller part.

Find the roots of the quadratic equation, if they exist, using the quadratic formula of Shridhar Acharya.
$x^2+4x+5=0$

Find the roots of the quadratic equation, if they exist, using the quadratic formula of Shridhar Acharya.
$x+\frac{1}{x}=3, x\ne 0$

Find the roots of the quadratic equation, if they exist, using the quadratic formula of Shridhar Acharya.
$9x^2+7x-2=0$

Solve the quadratic equation by the method of perfect the square.
$4x^2+4bx-\left(a^2-b^2\right)=0$

Solve the quadratic equation by the method of perfect the square.
$2x^2+x-4=0$

Solve the quadratic equation by the method of perfect the square.
$4x^2+4\sqrt{3}x+3=0$

Solve the quadratic equation by the method of perfect the square.
$5x^2-6x-2=0$

Solve the quadratic equation by the method of perfect the square.
$3x^2-5x+2=0$

Solve the equation by factorisation method.
$ab{x}^2+\left(b^2-ac\right)x-bc=0$

Solve the equation by factorisation method.

$4x^2-4a^{2}x+\left(a^4-b^4\right)=0$

Solve the equation by factorisation method.
$\frac{x+3}{x+2}=\frac{3x-7}{2x-3}$

Solve the equation by factorisation method.
$x^2+8x+7$

Solve the equation by factorisation method.
$3x^2-2\sqrt{6}x+2=0$

Solve the equation by factorisation method.
$100x^2-20x+1=0$