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

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

Without solving the equation

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

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

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

(b) Hence, or otherwise, write down

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

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

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

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

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

(b) show that $\displaystyle k = 5$

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

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

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

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

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

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

(a) Find the value of $\displaystyle p$

(b) Show that

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

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

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

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

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

(b) Hence, or otherwise, find

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(a) show that $\displaystyle k = -7$

(b) Hence form a quadratic equation that has roots

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

Without solving the equation,

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

The quadratic equation

has equal roots.

Find the two possible values of $\displaystyle k$

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

(b) Find

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

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

(c) Hence, or otherwise, write down

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

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

One solution to the following simultaneous equations

is $\displaystyle \left(-\frac{1}{2}, q\right)$, where $\displaystyle p$ is an integer and $\displaystyle q$ is a prime number.

(a) Find the value of $\displaystyle p$ and the value of $\displaystyle q$

(b) Hence find the other solution to the equations.

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

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

(a) show that

(b) Without solving $\displaystyle E$

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

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

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

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

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

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

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

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

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

(a) find the value of $\displaystyle p$

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

(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. $\displaystyle AEFG$ and $\displaystyle DLMN$ are squares with sides of length $\displaystyle 3x\,\mathrm{cm}$. $\displaystyle BPQR$ and $\displaystyle CTUV$ are rectangles with sides of length $\displaystyle 5x\,\mathrm{cm}$ and $\displaystyle 3x\,\mathrm{cm}$. The width of the cross is $\displaystyle y\,\mathrm{cm}$.

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

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

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

(b) show that

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

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

Figure 2 shows a rectangle with width $\displaystyle x$ metres and length $\displaystyle (x+4)$ metres. Its perimeter is $\displaystyle P$ metres and its area is $\displaystyle A\,\mathrm{m}^2$.

(a) Find, in terms of $\displaystyle x$, an expression for

(i) $\displaystyle P$,

(ii) $\displaystyle A$.

The perimeter has to be less than $\displaystyle 30$ metres. The area has to be greater than $\displaystyle 12\,\mathrm{m}^2$.

(b) Find the set of possible values for $\displaystyle x$. Give your answer in the form

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

has real roots.

The roots of the quadratic equation

are $\displaystyle \alpha$ and $\displaystyle \beta$.

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

(i) $\displaystyle \alpha+\beta$,

(ii) $\displaystyle \alpha\beta$.

(b) Show that

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

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

The quadratic equation

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

(a) Show that

The quadratic equation

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

(b) Show that

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

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

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

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

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

(b) Hence write down

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

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

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

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

(b) Hence, or otherwise, find

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

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

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

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

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

(a) Express

in the form $\displaystyle (x+a)^2+b$, where $\displaystyle a$ and $\displaystyle b$ 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 $\displaystyle y=x^2+4x-8$ and the line $\displaystyle y=2x+7$, showing clearly the coordinates of the turning point and the points of intersection.

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

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

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

(a) show that

(b) find the value of $\displaystyle q$.

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

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

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

where $\displaystyle p$ and $\displaystyle q$ are constants.

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

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

The curve $\displaystyle C$ with equation

crosses the $\displaystyle x$-axis at the points $\displaystyle A$ and $\displaystyle B$, as shown in Figure 3.

(a) Find the coordinates of $\displaystyle A$ and $\displaystyle B$.

The tangent to $\displaystyle C$ at $\displaystyle A$ meets the tangent to $\displaystyle C$ at $\displaystyle B$ at $\displaystyle T$.

(b) Find the coordinates of $\displaystyle T$.

The normal to $\displaystyle C$ at $\displaystyle A$ meets the normal to $\displaystyle C$ at $\displaystyle B$ at $\displaystyle N$.

(c) Find the coordinates of $\displaystyle N$.

(d) Find the area of quadrilateral $\displaystyle ATBN$.

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

has real roots.

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

(b) form a quadratic equation with roots

Give each coefficient in terms of $\displaystyle k$.

The function $\displaystyle f$ is defined by

Given that

where $\displaystyle a$, $\displaystyle b$ and $\displaystyle c$ are constants,

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

(b) Hence write down

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

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

Given that $\displaystyle f(x)$ can be written in the form

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

(b) Write down

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

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

The equation

has unequal real roots.

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

The roots of the equation

are $\displaystyle \alpha$ and $\displaystyle \beta$.

(a) Without solving the equation, find

(i) the value of $\displaystyle \alpha^2+\beta^2$,

(ii) the value of $\displaystyle \alpha^4+\beta^4$.

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

(b) show that

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

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

Given that

where $\displaystyle p$ and $\displaystyle q$ are positive constants,

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

The roots of the equation

are $\displaystyle \alpha$ and $\displaystyle \beta$, where $\displaystyle \alpha>\beta$.

Without solving the equation,

(a) find the value of $\displaystyle \alpha^2+\beta^2$.

(b) Show that

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

The quadratic equation

where $\displaystyle p>0$, has roots $\displaystyle \alpha$ and $\displaystyle \beta$.

Given that

and that

where $\displaystyle k>3$,

(a)

(i) write down the value of $\displaystyle q$,

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

Given also that

(b) find the value of $\displaystyle k$.

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