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Solve the quadratic equation:

$x^2 - 9 = 0$

Try using different methods, e.g. factorisation or completing the square as well as the quadratic formula.

Enter the lowest value first

$x=$ [[0]] , [[1]]

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Solve the quadratic equation:

$x^2 - 25 = 0$

Try using different methods, e.g. factorisation or completing the square as well as the quadratic formula.

Enter the lowest value first

$x=$ [[0]] , [[1]]

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Solve the quadratic equation:

$3x^2 - 12 = 0$

Try using different methods, e.g. factorisation or completing the square as well as the quadratic formula.

Enter the lowest value first

$x=$ [[0]] , [[1]]

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Solve the quadratic equation:

$x^2 - 5x + 6 = 0$

Try using different methods, e.g. factorisation or completing the square as well as the quadratic formula.

Enter the lowest value first

$x=$ [[0]] , [[1]]

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Solve the quadratic equation:

$x^2 +7x = 0$

Try using different methods, e.g. factorisation or completing the square as well as the quadratic formula.

Enter the lowest value first

$x=$ [[0]] , [[1]]

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Remember the square root of a negative number gives the imaginary number, i.

A complex number consists of a real part and an imaginary part.

Example,

$5x^2 - 6x + 5$
$x=\\frac{6\\pm\\sqrt{36-100}}{10}$
$x=\\frac{6\\pm\\sqrt{-64}}{10}$
$x=\\frac{6\\pm 8i}{10}$
$x=0.6+0.8i \\text{ and } 0.6-0.8i$

A complex number in Cartesian form: $z = a + bi$ can be expressed in polar form: $z = r(\\cos{\\theta} + i\\sin{\\theta})$ using the conversion:

$r^2 = a^2 + b^2$

$\\theta = \\tan^{-1} \\left(\\frac{b}{a}\\right)$

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Solve the following quadratic equation giving your answer in Cartesian and polar form

$x^2 - 6x + 25$

$x = $ [[0]] $\\pm$ [[1]] $i$

$x = $ [[2]] $(\\cos$ [[3]] + $i \\sin$ [[4]]$)$ (in degrees to 2 d.p.)

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General rules for partial fractions:

$\\frac{P(x)}{(ax+b)(cx+d)}=\\frac{A}{ax+b}+\\frac{B}{cx+d}$
$\\frac{P(x)}{(ax+b)^2}=\\frac{A}{ax+b}+\\frac{B}{(ax+b)^2}$
$\\frac{P(x)}{(ax^2+bx+c)(dx+e)}=\\frac{Ax+B}{ax^2+bx+c}+\\frac{C}{dx+e}$

Find the partial fraction of:

$\\frac{5x - 1}{(x + 1)(x - 2)}$

$ = [[0]]/$(x + 1)$ + [[1]]/$(x - 2)$

General rules for partial fractions:

$\\frac{P(x)}{(ax+b)(cx+d)}=\\frac{A}{ax+b}+\\frac{B}{cx+d}$
$\\frac{P(x)}{(ax+b)^2}=\\frac{A}{ax+b}+\\frac{B}{(ax+b)^2}$
$\\frac{P(x)}{(ax^2+bx+c)(dx+e)}=\\frac{Ax+B}{ax^2+bx+c}+\\frac{C}{dx+e}$

Find the partial fraction of:

$\\frac{3x + 14}{x^2 + 8x + 16}$

$ = [[0]]/$(x+4)$ + [[1]]/$(x+4)^2$

General rules for partial fractions:

$\\frac{P(x)}{(ax+b)(cx+d)}=\\frac{A}{ax+b}+\\frac{B}{cx+d}$
$\\frac{P(x)}{(ax+b)^2}=\\frac{A}{ax+b}+\\frac{B}{(ax+b)^2}$
$\\frac{P(x)}{(ax^2+bx+c)(dx+e)}=\\frac{Ax+B}{ax^2+bx+c}+\\frac{C}{dx+e}$

Find the partial fraction of:

$\\frac{6x^2 + 13x + 2}{(x^2 + 5x + 1)(x-1)}$

$ = ([[0]]$x$ + [[2]])/$(x^2 + 5x + 1)$ + [[1]]/$(x-1)^2$

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Example,

$\\displaystyle \\int 2xe^x\\,dx$
$u = 2x \\rightarrow \\frac{du}{dx}=2$
$\\frac{dv}{dx}=e^x \\rightarrow v = \\int e^x.dx = e^x$
$\\int 2xe^x.dx = (2x)\\left(e^x\\right) - \\int \\left(e^x \\right) \\left(2\\right).dx$
$=2xe^x - 2\\int e^x.dx$
$=2xe^x - 2e^x + C$
$=2e^x(x-1) + C$

The tabular method could also be used. In this approach, one function is differentiated until it reaches zero. The other is integrated the corresponding number of times. An alternative sign approach is then used to multiple down the table and form the answer. Please see the notes from AMR129 for an example of this.

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Solve the following integral using the 'by parts' method:

$\\int x\\sin{(2x)}\\, dx$

Enter your answer in the same way you would in MATLAB.

Fractions can be entered using the \\ symbol.

Variables must be multiplied using the * symbol, for example $xy$ should be entered as x*y

Functions must include brackets, for example $\\sin{x}$ should be entered as sin(x)

Don't forget to include the \"+ C\" as it is an indefinite integral.

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Example,

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Solve the following integral using the 'by parts' method:

$\\int xe^{3x}\\, dx$

Enter your answer in the same way you would in MATLAB.

Fractions can be entered using the \\ symbol.

Variables must be multiplied using the * symbol, for example $xy$ should be entered as x*y

Functions must include brackets, for example $\\sin{x}$ should be entered as sin(x)

Powers can be entered using the ^ symbol, for example $e^x$ can be entered as e^(x)

Don't forget to include the \"+ C\" as it is an indefinite integral.

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Solve the following integral using the 'by parts' method:

$\\int x^2\\cos{(kx)}\\, dx$

Enter your answer in the same way you would in MATLAB.

Fractions can be entered using the \\ symbol.

Variables must be multiplied using the * symbol, for example $xy$ should be entered as x*y

Functions must include brackets, for example $\\sin{x}$ should be entered as sin(x)

Powers can be entered using the ^ symbol, for example $e^x$ can be entered as e^(x)

Don't forget to include the \"+ C\" as it is an indefinite integral.

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