The roots of the equation below are real numbers.

$2x^2-20x+c=0$

For what value of $c,$ the roots will be equal$?$ Show your steps and give valid reasons.

When a marble is dropped from an initial height, $d$ metres, with an initial speed, $v$ m/s, the height of the marble at time $t$ is represented by $h\left(t\right)=vt-2t^2+d.$

A marble is dropped from a height of $48$ m with an initial speed, $10$ m/s. How long does it take for the marble to hit the ground$?$ Show your steps and give valid reasons.

Find the solution(s) of the following equation.

$\left(y-1\right)\left(y-3\right)\left(\frac{1}{y-3}+\frac{2y}{y-1}\right)=2, y\ne 1,3$

Show your steps and give valid reasons.

The graph below represents the path of a ball thrown by Ankush. The maximum height, $h,$ the ball reaches with respect to time, $t,$ is represented by the polynomial $h\left(t\right)=-t^2+\frac{19}{4}t+\frac{5}{4}.$

How long will it take for the ball to reach the ground$?$ Show your steps.

Shown below is the graph of a quadratic polynomial, $y\left(x\right)=p{x}^2+qx+r.$

Find the nature of the roots of the quadratic equation, $p{x}^2+qx+r=0.$ Give a reason for your answer.

${\left(2x+1\right)}^3=8x\left(x^2+1\right)+3$

Write the zeros of the above equation in the form $\left(x,y\right).$ Show your steps. 

$3{\left(3\right)}^{2m}+11{\left(3\right)}^m=4$

Use the substitution $3^m=x$ to solve for $m.$ Show your steps.

Three students were asked how they would verify their solution of a quadratic equation, $\left(x-2\right)\left(x-5\right)=0.$ Shown below are their responses.

Student $1$ said, "In the first bracket, $x$ must equal $2,$ and in the second bracket, $x$ must equal $5.$ So $\left(2-2\right)\left(5-5\right)=0.$"

Student $2$ said, "In the first bracket,$x$ must equal $2$, but in the second bracket, $x$ can have any real number value. For example $\left(2-2\right)\left(3-5\right)=0$ or $\left(2-2\right)\left(10-5\right)=0.$"

Student $3$ said, "Both brackets should always have the same $x$ value. So, $x$ is either $2$ or $5$ in both brackets. For example, $\left(2-2\right)\left(2-5\right)=0$ and $\left(5-2\right)\left(5-5\right)=0.$"

Whose response is correct$?$

Aman solved a quadratic equation and found its roots to be real.

Which of these could represent the graph of the equation Aman solved$?$