Quadratics (Edexcel AS Maths: Pure): Exam Questions

A function is defined by

$\mathrm{g}\left(x\right)=12+4x-x^2$

The curve $y=\mathrm{g}\left(x\right)$ meets the $y$-axis at the point $P$.

(i) Find all values of $x$ for which $\mathrm{g}\left(x\right)=0$.

(ii) Write down the coordinates of $P$.

(i) Express $\mathrm{g}\left(x\right)$ in the form $a-(x-b{)}^2$, where $a$ and $b$ are constants to be found.

(ii) Hence write down the coordinates of the turning point on the graph of $y=\mathrm{g}\left(x\right)$.

Sketch the graph of $y=\mathrm{g}\left(x\right)$.

Label clearly the coordinates of

• any turning points

• any points where the graph meets the coordinate axes

A function is given by

$\mathrm{f}\left(x\right)=x^2+4x+5$

By expressing the function in the form

$\mathrm{f}\left(x\right)={\left(x+a\right)}^2+b$

where $a$ and $b$ are integers to be found, find

(i) the minimum value of the function

(ii) the value of $x$ for which the function is at its minimum value.

Hence prove that the equation

$\mathrm{f}\left(x\right)=0$

 has no real roots.

A function is given by

$\mathrm{f}\left(x\right)=k{x}^2+2kx-3$

The equation $\mathrm{f}\left(x\right)=0$ has two distinct real roots.

Find the possible values of $k$.

The equation

$2x^2-4x+3-2k=0$

has real roots.

Find the possible values of $k$.

The equation

$x^2+px+q=0$

has no real roots.

Show that

$p^2<4q$

Solve the equation

$x^4-13x^2+36=0$

Solve the equation

$x^{\frac{2}{5}}+x^{\frac{1}{5}}=6$

A function is defined by

$\mathrm{f}\left(x\right)=(ax-b{)}^2$

where $a$ and $b$ are non-zero constants.

A teacher claims that the quadratic expression must have a discriminant of zero.

Without expanding the brackets, explain why this must be true.

Express

$y=4x^2+8x-5$

 in the form

$y=a(x+b{)}^2+c$

where $a$, $b$ and $c$ are integers to be found.

Hence write down the coordinates of the minimum point on the curve.

Sketch the graph of $y=4x^2+8x-5$.

Label clearly the coordinates of

• any turning points

• any points where the graph meets the coordinate axes

Find the solutions to the equation

$2x^2+x-6=0$

A curve has the equation

$y=2x^2+x-6$

By expressing the equation of the curve in the form

$y=a{\left(x+b\right)}^2+c$

where $a$, $b$ and $c$ are constants to be found, find the coordinates of the turning point on the curve.

Sketch the graph of $y=2x^2+x-6$

Label clearly the coordinates of

• any turning points

• any points where the graph meets the coordinate axes

Sketch the graph of $y=12x^2-5x-72$.

Label clearly the coordinates of any points where the curve meets the coordinate axes. 

The diagram below shows the graph of $y=\mathrm{f}\left(x\right)$.

The coordinates of the turning point and the points where the graph meets the coordinate axes have been labelled.

Sketch the graph of $y=\mathrm{f}(x+3)$.

Label clearly the coordinates of

• any turning points

• any points where the graph meets the coordinate axes

The figure below shows the curve $y=\mathrm{f}\left(x\right)$.

The coordinates of any points where the curve meets the coordinate axes are shown.

The coordinates of the maximum point are also shown.

The graph of $y=\mathrm{f}\left(x\right)+6$ meets the $x$-axis at two points, one of which has coordinates $\left(-2,0\right)$.

Sketch the graph of $y=\mathrm{f}\left(x\right)+6$.

Label clearly the coordinates of

• the maximum point on the curve

• any points where the curve meets the coordinate axes

A curve $C$ has equation $y=\mathrm{f}\left(x\right)$ where

$\mathrm{f}\left(x\right)=–3x^2+12x+8$

Write $\mathrm{f}\left(x\right)$ in the form

$a{\left(x+b\right)}^2+c$

where $a$, $b$ and $c$ are constants to be found.

The curve $C$ has a maximum turning point at $M$.

Find the coordinates of $M$.

$\begin{array}{ccc}\mathrm{f}\left(x\right)=2x^2+4x+9& & x\in \mathrm{ℝ}\end{array}$

Write $\mathrm{f}\left(x\right)$ in the form $a{\left(x+b\right)}^2+c$, where $a,b$ and $c$ are integers to be found.

Sketch the curve with equation $y=\mathrm{f}\left(x\right)$ showing any points of intersection with the coordinate axes and the coordinates of any turning point.

The equation $0=k{x}^2+2kx-3$ has two distinct real roots.

The equation $0=k{x}^2+4kx-16$ has no real roots.

Find the possible values of $k$.

The curve $C$ has the equation $y=x^2-3x+2$

The line $l$ has the equation $y=3x-7$

On the same coordinate axes, sketch $C$ and $l$.

Label clearly the coordinates of

• any points of intersection between $C$ and $l$

• all points where $l$ meets the coordinate axes

• all points where $C$ meets the coordinate axes

• any turning points on $C$

The curve with equation

$y=3x^2+2px+4q$

where $p$ and $q$ are non-zero constants does not meet the $x$-axis.

Show that 

$p^2<kq$

where $k$ is a constant to be found.

Given that the curve

$y=3x^2+2px+4q$

passes through the points with coordinates $(-2,6)$ and $(2,6)$, find the values of $p$ and $q$.

The equation

$2k-3kx-x^2=0$

where $k$ is a negative constant has has two distinct real roots.

Find the possible values of $k$.

In the case where $k=-1$, sketch the graph of $y=2k-3kx-x^2$.

Label clearly the coordinates of all points where the graph meets the coordinate axes.

A function is given by

$\mathrm{f}\left(x\right)=x^2+8x+c$

where $c$ is a constant.

By expressing the function in the form

$\mathrm{f}\left(x\right)={\left(x+a\right)}^2+b$

find, in terms of $c$ where necessary,

(i) the minimum value of the function

(ii) the value of $x$ for which the function is at its minimum value

Find the possible values of $c$ for which the equation $\mathrm{f}\left(x\right)=0$ has no real roots.

A model for the arch of a bridge over a river is given by

$y=4-\frac{x^2}{8}$

The minimum water level of the river under the bridge is represented by the $x$-axis and all measurements are in metres.

The width of the river at its minimum water level is the distance between the two $x$-intercepts.

The maximum water level is given by the line $y=0.5$

Determine whether the width of the river under the bridge at its maximum water level exceeds 11 metres.

A canal boat is modelled as a cuboid above water level with a cross-section measuring 6 m wide and 2.5 m tall.

Determine whether it is possible for the canal boat to fit underneath the bridge when the river is at its minimum water level.

To support the bridge, the arch will continue 2.5 m vertically below the minimum water level.

Find the exact distance between the ends of the base of the arch.

Solve the equation

$2^{2x}+64=20\left(2^x\right)$

Solve the equation 

$5\sqrt{x}+3=2x$

Solve the equation

$x^{\frac{2}{3}}+2x^{\frac{1}{3}}=8$

A stone is thrown vertically upwards from the top of a cliff. The stone lands in the sea vertically below. 

The path of the stone is modelled by

$\mathrm{h}\left(t\right)=24+2t-0.5t^{2}t\ge 0$

where

• $\mathrm{h}$ is the height, in metres, of the stone above sea level

• $t$ is the time in seconds since the stone was thrown

Write down the vertical height above sea level from which the stone was thrown.

Find the maximum height the stone reaches above sea level.

Find how long it takes for the stone to reach the sea.

Give your answer in seconds to 1 decimal place.

Express 

$y=-6x^2+8x-5$

in the form 

$y=a-b{(x+c)}^2$

where $a$, $b$ and $c$ are constants to be found.

Hence write down the exact coordinates of the maximum point on the curve with equation

$y=-6x^2+8x-5$

Sketch the graph of

$y=-6x^2+8x-5$

Label clearly the coordinates of

• the maximum point of the curve

• any points where the curve meets the coordinate axes

The minimum point on the curve with equation 

$y=x^2+px+q$

has coordinates $\left(3,1\right)$.

Find the values of $p$ and $q$.

The equation

$4k-6kx-x^2=0$

where $k<0$, has two distinct real roots, $\alpha$ and $\beta$, where $0<\alpha <\beta$.

Sketch the graph of $y=4k-6kx-x^2$.

Label clearly the points where the graph meets the coordinate axes.

Find the possible values of $k$.

Solve the equation

$8\sqrt{x}=48-x$

Solve the equation

$2^{4x}+64=20\left(2^{2x}\right)$

Given that

$x^2+6xy+9y^2=0$

find and simplify a relationship between $x$ and $y$.

A circle $C$has equation

$x^2+y^2+6kx-2ky+7=0$

where $k$ is a constant.

The line with equation$y=2x–1$ intersects $C$ at $2$ distinct points.

Find the range of possible values of $k$.

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

Using the substitution $u=\sqrt{x}$ or otherwise, solve

$6x+7\sqrt{x}-20=0$

In this question you should show all stages of your working.

Solutions relying on calculator technology are not acceptable.

Using algebra, find all solutions of the equation

$3x^3-17x^2-6x=0$

Hence find all real solutions of

$3{\left(y-2\right)}^6-17{\left(y-2\right)}^4-6{\left(y-2\right)}^2=0$

The functions $\mathrm{f}\left(x\right)$ and $\mathrm{g}\left(x\right)$ are defined by

$\begin{array}{rcl}\mathrm{f}\left(x\right)& =& (k-1)x^2-(k-2)x-2kx\in \mathrm{ℝ}\\ \mathrm{g}\left(x\right)& =& (k-1)x^2-3kx+k+1x\in \mathrm{ℝ}\end{array}$

 where $k$ is a non-zero constant and $k^2\ne 1$.

When the curve $y=\mathrm{f}\left(x\right)$ and the curve $y=\mathrm{g}\left(x\right)$ are plotted on the same set of coordinate axes, they intersect once only.

Find, in terms of $k$, the $x$-coordinate of the point of intersection.

Hence, in the case when $k=3$, find the exact coordinates of the point of intersection.

The equation

$k^2{x}^2-4x+5=k^2$

has two distinct real roots.

Find the possible values of $k$.

Show that the equation 

$a{x}^2+bx+c=0$

can be written in the form

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

where $a$, $b$ and $c$ are constants and $a\ne 0$.

Hence show that

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$