Quadratic Equations

If one root of the quadratic equation $a{x}^2+bx+c=0$ is equal to nth power of the other root, then the value of ${\left(a{c}^n\right)}^{\frac{1}{n+1}}+{\left(a^{n}c\right)}^{\frac{1}{n+1}}$ is equal to

The height of the handrail of a small bridge can be modelled by the equation $f\left(x\right)=0.267x-0.00889x^2$ where $f\left(x\right)$ is the height of the handrail above the walkway in metres, and $x$ is the horizontal distance from the start of the bridge in metres. The local authority wants to put a horizontal line of decorative lights at a height of $1$ metre above the walkway. The two ends of the lights will be attached to the handrail. Find the coordinates of the points where the lights are attached to the handrail and the lengths of the lights needed. (Write the answer in metres and up to one decimal place.)

The height of the handrail of a small bridge can be modelled by the equation $f\left(x\right)=0.267x-0.00889x^2$ where $f\left(x\right)$ is the height of the handrail above the walkway in metres, and $x$ is the horizontal distance from the start of the bridge in metres. The local authority wants to put a horizontal line of decorative lights at a height of $1$ metre above the walkway. The two ends of the lights will be attached to the handrail. Find the coordinates of the points where the lights are attached to the handrail.

An open cylinder has a volume of $400 {\mathrm{cm}}^3$. The radius of the base is $r \mathrm{cm}$ and the height is $h \mathrm{cm}$. Volume of cylinder is $\pi r^{2}h=400$. Rearrange the equation to make $h$ the subject.

The perimeter of picture is $400 \mathrm{cm}$. The length of the picture is $x \mathrm{cm}$. Find an equation for the area, $A {\mathrm{cm}}^2$, of the picture.

For the equation $f\left(x\right)=x^2+6x-7$ find the values of $a,b$ and $c$.

Anmol throws a stone in the air. The height of the stone, $h\left(x\right)$ metres, at time $x$ seconds is modelled by the equation $h\left(x\right)=-2.2625x^2+8.575x+1.9$

Find the time when the stone lands back on the ground. (Write the answer in seconds)

A company's profits, in $\mathrm{US}\$$, are modelled by the equation $f\left(x\right)=-0.9x^2+52x-360$ where $x$ is the number of units sold each week.

Find the number of units sold to make a profit of $\mathrm{US}\$300$ per week.

The path of an arrow can be modelled by the quadratic function $f\left(t\right)=-4.34t^2+25.77t+1.8$ where $t$ is the time in seconds and $f\left(t\right)$ is the height in metres. Find the times when the height is $10$ metres.

The path of a ball is modelled by the equation $f\left(x\right)=-3.53x^2+13.7x+1.75$ where $f\left(x\right)$ is the height of the ball in metres and $x$ is the time in seconds. Find the times when the height of the ball is $8$ metres.

The graph of $y=a{x}^2+bx+c$ passes through the points $\left(0,-9\right),$ and $\left(-1,9\right)$. Given that the equation of symmetry of the graph is $x=4$, find the values of $a,b,$ and $c$.

Given a general quadratic function of the form $f\left(x\right)=a{x}^2+bx+c$, write down the values of $a,b$, and $c$ for the following function.

$f\left(x\right)=-10+x+5x^2$

Given a general quadratic function of the form $f\left(x\right)=a{x}^2+bx+c$, write down the values of $a,b$, and $c$ for the following function.

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

Given a general quadratic function of the form $f\left(x\right)=a{x}^2+bx+c$, write down the values of $a,b$, and $c$ for the following function.

$f\left(x\right)=-2x^2+x+1$

Given a general quadratic function of the form $f\left(x\right)=a{x}^2+bx+c$, write down the values of $a,b$, and $c$ for the following function.

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

Given a general quadratic function of the form $f\left(x\right)=a{x}^2+bx+c$, write down the values of $a,b$, and $c$ for the following function.

$f\left(x\right)=x^2-4x+2$

Solution to the trigonometric equation ${\sin }^{2}x – 5\sin x + 4 = 0$ is 

Find all real numbers satisfying $4^x-3\left(2^{x+2}\right)+2^5=0$.

If $\alpha ,\beta$ are the roots of the equation $x^2-7x-4=0$, then ${\alpha }^3=53\alpha +k$. Find the value of $k$?

Solve the equation $\sqrt{3x+4}=x$.