Given that $f(x)$ can be written in the form $f(x) = a(x+b)^2 + c$ where $a$, $b$ and $c$ are rational numbers,

(a) find the value of $a$, the value of $b$ and the value of $c$

(b) Hence, or otherwise, write down

(i) the minimum value of $f(x)$

(ii) the value of $x$ at which this minimum value occurs.

(a) Show that $(\alpha + \beta)(\alpha^2 - \alpha\beta + \beta^2) = \alpha^3 + \beta^3$

The equation $2x^2 + 8x - k = 0$ has roots $\alpha$ and $\beta$ and where $k$ is a constant

Given that $\alpha^3 + \beta^3 = -94$

(b) show that $k = 5$

Given that $\alpha > \beta$ and without solving the equation $2x^2 + 8x - k = 0$

(c) (i) show that $\alpha - \beta = \sqrt{26}$

(ii) hence find the exact value of $\alpha^3 - \beta^3$

(a) Show that $(\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta) = \alpha^3 + \beta^3$

The quadratic equation $2x^2 - 6x - 7 = 0$ has roots $\alpha$ and $\beta$

Without solving the equation

(b) form a quadratic equation, with integer coefficients, which has roots $\frac{\alpha^2}{\beta}$ and $\frac{\beta^2}{\alpha}$

The curve $C$ has equation $y = f(x)$

The line $l$ with equation $y - 8x + 9 = 0$ is a tangent to $C$

The line $k$ is perpendicular to $l$ and is also the tangent to $C$ at the point where $x = p$

(a) Find the value of $p$

(b) Show that

The quadratic equation $f(x) = 0$ has roots $\alpha$ and $\beta$

Without solving the equation and using your results from part (b)

(c) form a quadratic equation with integer coefficients, that has roots

Given that $f(x)$ can be expressed in the form $A - B(x + C)^2$ where $A$, $B$ and $C$ are positive constants

(a) find the value of $A$, the value of $B$ and the value of $C$

(b) Hence write down the maximum value of $f(x)$

The equation $f(x) = 0$ has roots $\alpha$ and $\beta$

Without solving the equation $f(x) = 0$

(c) form a quadratic equation, with integer coefficients, that has roots $\frac{3\alpha}{\beta}$ and $\frac{3\beta}{\alpha}$

(d) Show that $(x + y)^3 = x^3 + y^3 + 3xy(x + y)$

$g(x) = 3x^2 + qx + r$ where $q$ and $r$ are constants

The equation $g(x) = 0$ has roots $\alpha^2 - \beta$ and $\beta^2 - \alpha$ where $\alpha$ and $\beta$ are the roots of the equation $f(x) = 0$

(e) Using your answer to part (d), find in simplified exact form, the value of $q$ and the value of $r$

Given that $f(x)$ can be written in the form $A(x + B)^2 + C$, where $A$, $B$ and $C$ are integers,

(a) find the value of $A$, the value of $B$ and the value of $C$

(b) Hence, or otherwise, find

(i) the value of $x$ for which $\frac{1}{f(x)}$ is a maximum

(ii) the maximum value of $\frac{1}{f(x)}$

The quadratic equation $2x^2 + kx + 4 = 0$ has roots $\alpha$ and $\beta$ such that

Given that $\alpha^2 - \beta^2 = \frac{7\sqrt{17}}{4}$

(a) show that $k = -7$

(b) Hence form a quadratic equation that has roots

The quadratic equation $3x^2 - 5x + 1 = 0$ has roots $\alpha$ and $\beta$

Without solving the equation,

form a quadratic equation with integer coefficients, that has roots $\frac{\alpha}{2\beta}$ and $\frac{\beta}{2\alpha}$

The quadratic equation

has equal roots.

Find the two possible values of $k$

(a) Express $g(x)$ in the form $p(x+q)^2 + r$ where $p, q$ and $r$ are rational numbers to be found.

(b) Find

(i) the minimum value of $g(x)$

(ii) the value of $x$ at which this minimum occurs.

(c) Hence, or otherwise, write down

(i) the minimum value of $h(x)$

(ii) the value of $x$ at which this minimum occurs.

The equation $kx^2 + 8x + 3k = 0$ where $k$ is a constant, has real unequal roots.

Find the set of values of $k$ giving your answer in an exact simplified form.

The roots of a quadratic equation are $\alpha$ and $\beta$ where

(a) Show that $\alpha\beta = 4$

(b) Form a quadratic equation with integer coefficients, that has roots

The roots of a quadratic equation $E$ are $\alpha$ and $\beta$ where $\alpha > \beta > 0$

Given that $\alpha - \beta = 2\sqrt{6}$ and $\alpha^2 + \beta^2 = 30$

(a) show that

(b) Without solving $E$

Given that $\alpha^4 = P + Q\sqrt{6}$ where $P$ and $Q$ are positive integers,

(c) find the value of $P$ and the value of $Q$

Show that for all values of $k$, the equation $f(x) = 0$ has distinct real roots.

The equation $f(x) = 0$ has roots $\alpha$ and $\beta$

(a) find the value of $p$

Without solving the equation $f(x) = 0$

(b) form a quadratic equation, with integer coefficients, that has roots

Figure 1 shows the design for a flag consisting of a white cross on a grey background. $AEFG$ and $DLMN$ are squares with sides of length $3x\,\mathrm{cm}$. $BPQR$ and $CTUV$ are rectangles with sides of length $5x\,\mathrm{cm}$ and $3x\,\mathrm{cm}$. The width of the cross is $y\,\mathrm{cm}$.

The total area of the flag is $H\,\mathrm{cm}^2$.

(a) Write down an expression, in terms of $x$ and $y$, for $H$.

Given that the area of the cross is $K\,\mathrm{cm}^2$,

(b) show that

The total area of the flag is to be $3432\,\mathrm{cm}^2$ and the area of the cross is to be $1080\,\mathrm{cm}^2$.

(c) Find the value of $x$ and the value of $y$.

Find the set of values of $k$ for which the equation

has real roots.

The roots of the quadratic equation

are $\alpha$ and $\beta$.

(a) Write down an expression in terms of $p$ for

(i) $\alpha+\beta$,

(ii) $\alpha\beta$.

(b) Show that

Given that $\alpha-\beta=3$,

(c) find the possible values of $p$.

The quadratic equation

has real roots. Find the set of possible values of $k$.

(a) Show that

The quadratic equation

where $k$ is a positive constant, has roots $\alpha$ and $\beta$, where $\alpha>\beta$.

(b) Show that

(c) Hence form a quadratic equation with roots $\alpha+1$ and $\beta-1$. Give your equation in the form

where $p$ and $q$ should be given in terms of $k$.

Given that $f(x)$ can be expressed in the form

where $A$, $B$ and $C$ are constants,

(a) find the value of $A$, the value of $B$ and the value of $C$.

(b) Hence write down

(i) the maximum value of $f(x)$,

(ii) the value of $x$ for which this maximum occurs.

Given that $f(x)$ can be written in the form $(x+a)^2+b$, where $a$ and $b$ are constants,

(a) find the value of $a$ and the value of $b$.

(b) Hence, or otherwise, find

(i) the minimum value of $f(x)$,

(ii) the value of $x$ for which this minimum occurs.

The curve $C$ has equation $y=f(x)$. The line $l$ has equation $y=x+5$.

(c) Use algebra to find the coordinates of the points of intersection of $C$ and $l$.

(d) Use algebraic integration to find the exact area of the finite region bounded by $C$ and $l$.

(a) Express

in the form $(x+a)^2+b$, where $a$ and $b$ are constants whose values are to be found.

(b) Use algebra to solve the simultaneous equations

Using the same axes and the results of parts (a) and (b),

(c) sketch the curve $y=x^2+4x-8$ and the line $y=2x+7$, showing clearly the coordinates of the turning point and the points of intersection.

The roots of the quadratic equation $f(x)=0$ are $\alpha$ and $\beta$.

The roots of the quadratic equation $g(x)=0$ are

Without solving the equation $f(x)=0$,

(a) show that

(b) find the value of $q$.

The equation $f(x)=0$ has roots $\alpha$ and $\beta$.

Without solving the equation $f(x)=0$,

(a) form an equation, with integer coefficients, that has roots

where $p$ and $q$ are constants.

Given that the equation $g(x)=0$ has roots $3\alpha+\beta$ and $\alpha+3\beta$,

(b) find the value of $p$ and the value of $q$.

(a) Find the set of values of $k$ for which the equation

has real roots.

Given that the roots of the equation are $\alpha$ and $\beta$,

(b) form a quadratic equation with roots

Give each coefficient in terms of $k$.

Given that $f(x)$ can be written in the form $a(x-b)^{2}+c$, where $a$, $b$ and $c$ are constants, find

(a) the value of $a$, the value of $b$ and the value of $c$.

(b) Hence write down

(i) the minimum value of $f(x)$,

(ii) the value of $x$ at which this minimum occurs.

Given that $f(x)$ can be written in the form $A(x+B)^{2}+C$,

(a) find the value of $A$, the value of $B$ and the value of $C$.

(b) Write down

(i) the minimum value of $f(x)$,

(ii) the value of $x$ at which this minimum occurs.

The equation $f(x)=px-6$ has unequal real roots.

(c) Find the set of possible values of $p$.

The roots of the equation $x^{2}+3x-5=0$ are $\alpha$ and $\beta$.

(a) Without solving the equation, find

(i) the value of $\alpha^{2}+\beta^{2}$

(ii) the value of $\alpha^{4}+\beta^{4}$

Given that $\alpha>\beta$ and without solving the equation

(b) show that $\alpha-\beta=\sqrt{29}$.

(c) Factorise $\alpha^{4}-\beta^{4}$ completely.

(d) Hence find the exact value of $\alpha^{4}-\beta^{4}$.

Given that $\beta^{4}=p+q\sqrt{29}$ where $p$ and $q$ are positive constants

(e) find the value of $p$ and the value of $q$.

The roots of the equation $3x^{2}-2x-1=0$ are $\alpha$ and $\beta$, where $\alpha>\beta$.

Without solving the equation,

(a) find the value of $\alpha^{2}+\beta^{2}$,

(b) show that $\alpha-\beta=\dfrac{4}{3}$,

(c) form a quadratic equation, with integer coefficients, that has roots $\dfrac{\alpha+\beta}{\alpha}$ and $\dfrac{\alpha-\beta}{\beta}$.

Given that $f(0)=-12$

(a) find the value of $q$.

(b) Find the range of values of $p$ for which the cubic equation $f(x)=0$ has only one real root.

The quadratic equation $x^2-px+q=0$ where $p>0$, has roots $\alpha$ and $\beta$.

Given that $2\alpha\beta=3$ and that $4(\alpha^2+\beta^2)=k^2-6k-3$ where $k>3$

(a) (i) write down the value of $q$,

(ii) find an expression, in terms of $k$, for $p$.

Given also that $7\alpha\beta=3(\alpha+\beta)$

(b) find the value of $k$.

(c) Hence form an equation, with integer coefficients, which has roots

Given that $f(x)$ can be written in the form $p(x+q)^2+r$, where $p$, $q$ and $r$ are rational numbers,

(a) find the value of $p$, the value of $q$ and the value of $r$. (3)

(b) Hence, or otherwise, find

(i) the maximum value of $f(x)$,

(ii) the value of $x$ for which this maximum occurs. (2)

(c) Write down

(i) the maximum value of $g(x)$,

(ii) the exact value of $x$ for which this maximum occurs. (3)

It is given that $\alpha$ and $\beta$ are such that $\alpha+\beta=-\dfrac52$ and $\alpha\beta=-5$.

(a) Form a quadratic equation with integer coefficients that has roots $\alpha$ and $\beta$. (2)

Without solving the equation found in part (a)

(b) find the value of

(i) $\alpha^2+\beta^2$

(ii) $\alpha^3+\beta^3$ (5)

(c) Hence form a quadratic equation with integer coefficients that has roots

The equation $3x^2-5x+4=0$ has roots $\alpha$ and $\beta$.

Without solving this equation, form a quadratic equation with integer coefficients that has roots

Here is a quadratic equation

where $p$ is a constant.

(a) Find the set of values of $p$ for which the equation has two real distinct roots. (5)

(b) List all the possible integer values of $p$ for which the equation has no real roots. (1)

The equation $3x^2-4x+6=0$ has roots $\alpha$ and $\beta$.

(a) Without solving the equation, write down

(i) the value of $\alpha+\beta$

(ii) the value of $\alpha\beta$ (2)

(b) Without solving the equation, show that $\alpha^3+\beta^3=-\frac{152}{27}$. (3)

(c) Form a quadratic equation, with integer coefficients, that has roots $\frac{\alpha}{\beta^2}$ and $\frac{\beta}{\alpha^2}$. (5)

Solve the equations

(a) Find the set of possible values of $p$ for which the equation $3x^2+px+3=0$ has no real roots. (3)

(b) Find the integer values of $q$ for which the equation $x^2+7x+q^2=0$ has real roots. (3)

$f(x)=x^2+px+7$, $p\in\mathbb R$.

The roots of the equation $f(x)=0$ are $\alpha$ and $\beta$.

(a) Find, in terms of $p$ where necessary,

(i) $\alpha^2+\beta^2\qquad$ (ii) $\alpha^2\beta^2$ (4)

Given that $7(\alpha^2+\beta^2)=5\alpha^2\beta^2$,

(b) find the possible values of $p$. (2)

Using the positive value of $p$ found in part (b) and without solving the equation $f(x)=0$,

(c) form a quadratic equation with roots $\displaystyle\frac{2p}{\alpha^2}$ and $\displaystyle\frac{2p}{\beta^2}$. (5)

The roots of the equation $f(x)=0$ are $\alpha$ and $\beta$.

(a) Without solving the equation $f(x)=0$, form an equation, with integer coefficients, which has

(i) roots $\dfrac\alpha\beta$ and $\dfrac\beta\alpha$ (6)

(ii) roots $2\alpha+\beta$ and $\alpha+2\beta$ (5)

(b) Express $f(x)$ in the form $A(x+B)^2+C$, stating the values of the constants $A,B$ and $C$. (3)

(c) Hence, or otherwise, show that the equation $f(x)=-8$ has no real roots. (2)

Given that $\alpha+\beta=5$ and $\alpha^2+\beta^2=19$

(a) show that $\alpha\beta=3$. (2)

(b) Hence form a quadratic equation, with integer coefficients, which has roots $\alpha$ and $\beta$. (2)

(c) Form a quadratic equation, with integer coefficients, which has roots $\dfrac\alpha\beta$ and $\dfrac\beta\alpha$. (5)

Given that $f(x)=l(x-m)^2+n$, for all values of $x$,

(a) find the value of $l$, the value of $m$ and the value of $n$. (3)

(b) Hence, or otherwise, find

(i) the minimum value of $f(x)$,

(ii) the value of $x$ for which this minimum occurs. (2)

(a) Show that $(\alpha+\beta)(\alpha^2-\alpha\beta+\beta^2)=\alpha^3+\beta^3$. (1)

The roots of the equation $2x^2+6x-7=0$ are $\alpha$ and $\beta$, where $\alpha>\beta$. Without solving the equation,

(b) find the value of $\alpha^3+\beta^3$. (4)

(c) show that $\alpha-\beta=\sqrt{23}$. (2)

(d) Hence find the exact value of $\alpha^3-\beta^3$. (2)

The equation $2x^2+3x+c=0$, where $c$ is a constant, has two equal roots.

(a) Find the value of $c$. (2)

(b) Solve the equation. (2)

The equation $2x^2+px-3=0$, where $p$ is a constant, has roots $\alpha$ and $\beta$.

(a) Find the value of

(i) $\alpha\beta$

(ii) $\left(\alpha+\dfrac1\beta\right)\left(\beta+\dfrac1\alpha\right)$ (4)

(b) Find, in terms of $p$,

(i) $\alpha+\beta$

(ii) $\left(\alpha+\dfrac1\beta\right)+\left(\beta+\dfrac1\alpha\right)$ (4)

Given that

(c) find the value of $p$. (1)

(d) Using the value of $p$ found in part (c), find a quadratic equation, with integer coefficients, which has roots $\alpha+\dfrac1\beta$ and $\beta+\dfrac1\alpha$. (2)