Quadratics

Find the set of values of $k$ for which the equation $2x^2 + 3kx + k = 0$ has two distinct real roots.

Feb/March 2017

Use an appropriate substitution to solve the equation $(2x - 3)^2 - \dfrac{4}{(2x - 3)^2} - 3 = 0$.

Feb/March 2021

The curve is defined by $y = x^2 + 2cx + 4$ and the straight line is given by $y = 4x + c$, where $c$ is a constant.

Feb/March 2022

The functions $f$, $g$ and $h$ are given by: $f : x \mapsto x - 4x^{\frac{1}{2}} + 1$ for $x \geq 0$, $g : x \mapsto mx^2 + n$ for $x \geq -2$, where $m$ and $n$ are constants, $h : x \mapsto x^{\frac{1}{2}} - 2$ for $x \geq 0$.

Feb/March 2022

Let $f$ be given by $f(x) = x^2 - 2x + 5$ for $x \in \mathbb{R}$. In the order shown, the graph of $y = f(x)$ is transformed to produce $y = g(x)$: A stretch parallel to the $x$-axis with scale factor $\frac{1}{2}$. A reflection across the $y$-axis. A stretch parallel to the $y$-axis with scale factor $3$.

Feb/March 2023

The curve is defined by $y = 5 + 3x - 2x^2$, and the straight line is defined by $y = kx + 13$, where $k$ is a constant.

Feb/March 2025

For $x \in \mathbb{R}$, the function $f$ maps $x$ to $2x^2 - 12x + 7$. For the same domain, $g$ maps $x$ to $2x + k$.

May/June 2010

The line $x - y + 4 = 0$ meets the curve $y = 2x^2 - 4x + 1$ at the points $P$ and $Q$. The coordinates of $P$ are given as $(3, 7)$.

May/June 2011

For the equation $x^2 + px + q = 0$, where $p$ and $q$ are constants, the roots are $-3$ and $5$.

May/June 2011

Determine the values of $m$ for which the line $y = mx + 4$ meets the curve $y = 3x^2 - 4x + 7$ at two separate points.

May/June 2011

Express $4x^2 - 12x$ in the shape $(2x + a)^2 + b$.

May/June 2014

Write $2x^2 - 10x + 8$ in the form $a(x + b)^2 + c$, where $a$, $b$ and $c$ are constants, and use your result to state the least value of $2x^2 - 10x + 8$.

May/June 2014

A wire of length $24$ cm is shaped to make the perimeter of a sector of a circle with radius $r$ cm.

May/June 2015

Write $2x^2 - 12x + 7$ in the form $a(x + b)^2 + c$, where $a$, $b$ and $c$ are constants.

May/June 2015

Determine the values of the constant $m$ for which the line $y = mx$ is tangent to the curve $y = 2x^2 - 4x + 8$.

May/June 2016

The curve with $\frac{dy}{dx} = 7 - x^2 - 6x$ goes through $(3, -10)$.

May/June 2017

The curve is defined by $y = x^2 - 6x + k$, with $k$ a constant.

May/June 2018

The function $f$ is defined as $f : x \mapsto 7 - 2x^2 - 12x$ for $x \in \mathbb{R}$.

May/June 2018

Rewrite $3x^2 - 12x + 7$ in the form $a(x + b)^2 + c$, where $a$, $b$ and $c$ are constants.

May/June 2018

The function $f$ is given by $f(x) = -2x^2 + 12x - 3$ for every $x \in \mathbb{R}$.

May/June 2019

The function $f$ is given by $f(x) = x^2 - 4x + 8$ for $x \in \mathbb{R}$.

May/June 2019

Find the values of $m$ for which the line $y = mx + 1$ and the curve $y = 3x^2 + 2x + 4$ meet at two distinct points.

May/June 2020

Write $16x^2 - 24x + 10$ in the form $(4x + a)^2 + b$.

May/June 2021

Write $x^2 - 8x + 11$ in the form $(x + p)^2 + q$, with $p$ and $q$ both constants.

May/June 2022

In the expansion of $(2x^2 + \frac{k^2}{x})^5$, let the coefficient of $x^4$ be $a$. In the expansion of $(2kx - 1)^4$, let the coefficient of $x^2$ be $b$.

May/June 2022

The function $f$ is defined by $f(x) = 4\cos^4 x + \cos^2 x - k$ for $0 \leq x \leq 2\pi$, where $k$ is a constant.

May/June 2022

In the expansion of $(p + \frac{1}{p}x)^4$, the coefficient of $x^3$ is 144.

May/June 2022

Write $4x^2 - 24x + p$ in the form $a(x + b)^2 + c$, with $a$ and $b$ integers and $c$ expressed using the constant $p$.

May/June 2023

Solve $8x^6 + 215x^3 - 27 = 0$.

May/June 2023

The function $f$ is given on $x \in \mathbb{R}$ by $f(x) = x^2 - 6x + c$, where $c$ is a constant. It is known that $f(x) > 2$ for every value of $x$.

May/June 2023

Write $3y^2 - 12y - 15$ in the form $3(y + a)^2 + b$, with $a$ and $b$ as constants.

May/June 2024

Use completing the square to determine the exact solutions of the equation $4x^2 - 4x - 1 = 0$.

May/June 2025

The curve is defined by $y = kx^2 + 1$ and the line is defined by $y = kx$, where $k$ is a non-zero constant.

Oct/Nov 2010

A diagram gives a layout for a rectangular park $ABCD$, where $AB = 40\,\text{m}$ and $AD = 60\,\text{m}$. The points $X$ and $Y$ are located on $BC$ and $CD$ respectively, and the paths $AX$, $XY$ and $YA$ enclose a triangular playground. The distance $DY$ is $x\,\text{m}$, while $XC$ is $2x\,\text{m}$.

Oct/Nov 2012

The curve is given by the equation $y = 2x^2 - 3x$.

Oct/Nov 2013

Solve for the values of $x$ in the inequality $x^2 - x - 2 > 0$.

Oct/Nov 2013

For $p \leq x \leq q$, where $p$ and $q$ are positive constants, the function $f$ is defined by $f : x \mapsto x^2 - 2x - 15$. Its range is stated as $c \leq f(x) \leq d$, where $c$ and $d$ are constants.

Oct/Nov 2014

Find the set of values of $k$ for which the line $y = 2x - k$ meets the curve $y = x^2 + kx - 2$ at two distinct points.

Oct/Nov 2014

For $x \in \mathbb{R}$, the function $f$ is given by $f: x \mapsto x^2 + ax + b$, with $a$ and $b$ as constants.

Oct/Nov 2015

The line is given by $y = 2x - 7$ and the curve is given by $y = x^2 - 4x + c$, with $c$ a constant. Find the set of values of $c$ for which the line does not intersect the curve.

Oct/Nov 2015

Write $x^2 + 6x + 2$ in the form $(x + a)^2 + b$, with $a$ and $b$ as constants.

Oct/Nov 2016

Determine the values of $k$ for which the curve $y = kx^2 - 3x$ and the line $y = x - k$ have no points in common.

Oct/Nov 2016

The curve has equation $y = -\frac{2}{x}$, while the straight line is given by $y = ax + 3a$.

Oct/Nov 2017

With all necessary working shown, solve the equation $4x - 11x^2 + 6 = 0$.

Oct/Nov 2018

The curve is given by $y = 2x + \frac{12}{x}$, while the line has equation $y + x = k$, where $k$ is a constant.

Oct/Nov 2018

The function $f$ is specified by $f : x \mapsto 2x^2 - 12x + 7$ for $x \in \mathbb{R}$.

Oct/Nov 2018

The curve is given by $y = 2x^2 - 3x + 1$ and the line is given by $y = kx + k^2$, where $k$ is a constant.

Oct/Nov 2018

A line is given by $y = 3kx - 2k$, and a curve is given by $y = x^2 - kx + 2$, where $k$ is a constant.

Oct/Nov 2019

Determine the set of values of $m$ for which the line $y = mx - 3$ and the curve $y = 2x^2 + 5$ do not meet.

Oct/Nov 2020

Write $x^2 + 6x + 5$ in the form $(x + a)^2 + b$, where $a$ and $b$ are constants.

Oct/Nov 2020

A curve is given by $y = 3x^2 - 4x + 4$, while a straight line is given by $y = mx + m - 1$, where $m$ is a constant.

Oct/Nov 2020

The curve is defined by $y = kx^2 + 2x - k$, while the line is defined by $y = kx - 2$, where $k$ is a constant.

Oct/Nov 2021

Solve for $x$ in $3x + 2 = \frac{2}{x - 1}$.

Oct/Nov 2022

In the expansion of $(1 + \frac{2}{p}x)^5 + (1 + px)^6$, the coefficient of $x^2$ is 70.

Oct/Nov 2022

The functions $f$ and $g$ are each defined for every $x \in \mathbb{R}$, and are given by $f(x) = x^2 - 4x + 9$ and $g(x) = 2x^2 + 4x + 12$.

Oct/Nov 2022

The curve is defined by the equation $y = 4x^2 + 20x + 6$.

Oct/Nov 2022

The function $f$ has the rule $f(x) = -2x^2 - 8x - 13$ for $x < -3$.

Oct/Nov 2022

The curve is given by the equation $y = x^2 - 8x + 5$.

Oct/Nov 2023

Determine the set of values of the constant $k$ for which the quadratic equation $3kx^2 + (k + 8)x + 3 = 0$ has two distinct real roots.

Oct/Nov 2025

Write $1-6x-x^2$ in the form $a-(x+b)^2$, with $a$ and $b$ as constants.

Oct/Nov 2025

Express $9x^2 - 36x + 8$ in the form $p(x + q)^2 + r$, where $p$, $q$ and $r$ are constants.

Oct/Nov 2025

The function $f$ has the rule $f(x) = px^2 + 4x + q$ for $x \in \mathbb{R}$, where $p$ and $q$ are constants.

Oct/Nov 2025

Write $4x^2 + 10x + 6$ in the form $a(x+b)^2 + c$, where $a$, $b$ and $c$ are rational constants that must be found.

Oct/Nov 2025

Solve the equation $x^3 - 28 + \frac{27}{x^3} = 0$ for $x$.

Oct/Nov 2025