Quadratics

Solve $\displaystyle x^{2}-11x+24=0.$

The diagram below shows the path of a small rocket which is fired into the air. The height, $\displaystyle h$ metres, of the rocket after $\displaystyle t$ seconds is given by $\displaystyle h(t)=16t-t^2.$

(a)  After how many seconds will the rocket first be at a height of 60 metres?

(b)  Will the rocket reach a height of 70 metres? Justify your answer.

Solve the equation by factorising: $\displaystyle 6x^2+13x-5=0$

Determine the nature of the roots of the function $\displaystyle f(x)=7x^{2}+5x-1$.

Determine the nature of the roots of the function $\displaystyle f(x)=4x^2+6x-1.$

Find the range of values of $\displaystyle p$ such that the equation $\displaystyle px^2-2x+3=0\small,\normalsize\ p\neq 0\small,$ has no real roots.

A storage unit, built in the shape of a cuboid, is shown.

It has length $\displaystyle (x+7)$ metres, breadth $\displaystyle x$ metres and height $\displaystyle 2$ metres.
The volume of this unit is $\displaystyle 45$ cubic metres.
(a)  Show that $\displaystyle 2x^2+14x-45=0$
(b)  Calculate $\displaystyle x,$ the breadth of the storage unit.
Give your answer correct to 1 decimal place.

Solve the equation $\displaystyle 3x^2+8x+1=0.$
Give your answers correct to 2 decimal places.

The diagram shows a parabola with equation of the form $\displaystyle y=(x+a)^{2}+b$.

(a)  State the value of $\displaystyle a.$
(b)  State the value of $\displaystyle b.$