Fractions, conjugates and rearranging

Hard0 of 21 parts

These are the parts of the section that calculus uses most heavily, and they are the ones worth over-practising. Several of them are exactly the manipulation a derivative asks for, with the calculus removed.

Simplify $\dfrac{x^{2}-9}{x^{2}-x-6}$ .

in terms of x

Write $\dfrac{2}{x+1} + \dfrac{3}{x-2}$ as a single fraction.

in terms of x

Simplify the complex fraction $\dfrac{\frac{1}{x}-\frac{1}{3}}{x-3}$ .

in terms of x

Rationalise the numerator of $\dfrac{\sqrt{x+4}-2}{x}$ .

in terms of x

Rationalise the numerator of $\dfrac{\sqrt{x}-\sqrt{7}}{x-7}$ .

in terms of x

Write $x^{1/2}+x^{-1/2}$ as a single fraction.

in terms of x

Write $2x(x+3)^{-1/2} + (x+3)^{1/2}$ as a single fraction.

in terms of x

Writing $x^{2}+10x+7$ in the form $(x+p)^{2}+q$ , what is $q$ ?

Writing $2x^{2}-12x+5$ in the form $2(x-h)^{2}+k$ , what is $k$ ?

Solve $A = P(1+rt)$ for $r$ .

in terms of A, P, t

Simplify $\dfrac{x^{2}-1}{x^{2}+4x+3}\cdot\dfrac{x+3}{x}$ .

in terms of x

Simplify $\dfrac{x^{2}-4}{x^{2}+6x+9} \div \dfrac{x-2}{x+3}$ .

in terms of x

Simplify $\dfrac{x^{2}-x-6}{x^{2}-4}\cdot\dfrac{x+2}{x-3}$ .

in terms of x

Rationalise the denominator of $\dfrac{3}{\sqrt{x}-\sqrt{2}}$ , for $x > 2$ .

in terms of x

Rationalise the numerator of $\dfrac{\sqrt{x+h}-\sqrt{x}}{h}$ .

in terms of x, h

Write $x^{-3/2}+x^{-1/2}$ as a single fraction.

in terms of x

Write $x^{2}+7x+3$ in the form $(x+p)^{2}+q$ .

in terms of x

Write $x^{2}-3x-5$ in the form $(x+p)^{2}+q$ .

in terms of x

Write $2x^{2}+10x+3$ in the form $a(x+p)^{2}+q$ .

in terms of x

Solve $\dfrac{1}{c} = \dfrac{1}{a}+\dfrac{1}{b}$ for $a$ .

in terms of b, c

Solve $v^{2} = u^{2}+2as$ for $a$ .

in terms of v, u, s