Solutions

Question 1

1. Simplify the following expressions:

\(\frac{3}{\sqrt{3}}\):

import sympy as sym

expression = sym.S(3) / sym.sqrt(3)
sym.simplify(expression)

\[\displaystyle \sqrt{3}\]

\(\frac{2 ^ {78}}{2 ^ {12}2^{-32}}\):

sym.S(2) ** 78 / (sym.S(2) ** 12 * sym.S(2) ** (-32))

\[\displaystyle 316912650057057350374175801344\]

\(8^0\):

sym.S(8) ** 0

\[\displaystyle 1\]

\(a^4b^{-2}+a^{3}b^{2}+a^{4}b^0\):

a = sym.Symbol("a")
b = sym.Symbol("b")
sym.factor(a ** 4 * b ** (-2) + a ** 3 * b ** 2 + a ** 4 * b ** 0)

\[\displaystyle \frac{a^{3} \left(a b^{2} + a + b^{4}\right)}{b^{2}}\]

Question 2

2. Solve the following equations:

\(x + 3 = -1\):

x = sym.Symbol("x")
equation = sym.Eq(x + 3, -1)
sym.solveset(equation, x)

\[\displaystyle \left\{-4\right\}\]

\(3 x ^ 2 - 2 x = 5\):

equation = sym.Eq(3 * x ** 2 - 2 * x, 5)
sym.solveset(equation, x)

\[\displaystyle \left\{-1, \frac{5}{3}\right\}\]

\(x (x - 1) (x + 3) = 0\):

equation = sym.Eq(x * (x - 1) * (x + 3), 0)
sym.solveset(equation, x)

\[\displaystyle \left\{-3, 0, 1\right\}\]

\(4 x ^3 + 7x - 24 = 1\):

equation = sym.Eq(4 * x ** 3 + 7 * x - 24, 1)
sym.solveset(equation, x)

\[\displaystyle \left\{- \frac{7}{12 \sqrt[3]{\frac{25}{8} + \frac{\sqrt{51654}}{72}}} + \sqrt[3]{\frac{25}{8} + \frac{\sqrt{51654}}{72}}, - \frac{\sqrt[3]{\frac{25}{8} + \frac{\sqrt{51654}}{72}}}{2} + \frac{7}{24 \sqrt[3]{\frac{25}{8} + \frac{\sqrt{51654}}{72}}} + i \left(\frac{7 \sqrt{3}}{24 \sqrt[3]{\frac{25}{8} + \frac{\sqrt{51654}}{72}}} + \frac{\sqrt{3} \sqrt[3]{\frac{25}{8} + \frac{\sqrt{51654}}{72}}}{2}\right), - \frac{\sqrt[3]{\frac{25}{8} + \frac{\sqrt{51654}}{72}}}{2} + \frac{7}{24 \sqrt[3]{\frac{25}{8} + \frac{\sqrt{51654}}{72}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{25}{8} + \frac{\sqrt{51654}}{72}}}{2} - \frac{7 \sqrt{3}}{24 \sqrt[3]{\frac{25}{8} + \frac{\sqrt{51654}}{72}}}\right)\right\}\]

Question 3

3. Consider the equation: \(x ^ 2 + 4 - y = \frac{1}{y}\):

Find the solution to this equation for \(x\).

y = sym.Symbol("y")
equation = sym.Eq(x ** 2 + 4 - y, 1 / y)
solution = sym.solveset(equation, x)
solution

\[\displaystyle \left\{- \sqrt{\frac{y^{2} - 4 y + 1}{y}}, \sqrt{\frac{y^{2} - 4 y + 1}{y}}\right\}\]

Obtain the specific solution when \(y = 5\). Do this in two ways: substitute the value in to your equation and substitute the value in to your solution.

solution.subs({y: 5})

\[\displaystyle \left\{- \frac{\sqrt{30}}{5}, \frac{\sqrt{30}}{5}\right\}\]

solution = sym.solveset(equation.subs({y: 5}), x)
solution

\[\displaystyle \left\{- \frac{\sqrt{30}}{5}, \frac{\sqrt{30}}{5}\right\}\]

Question 4

4. Consider the quadratic: \(f(x)=4x ^ 2 + 16x + 25\):

Calculate the discriminant of the quadratic equation \(4x ^ 2 + 16x + 25 = 0\). What does this tell us about the solutions to the equation? What does this tell us about the graph of \(f(x)\)?

quadratic = 4 * x ** 2 + 16 * x + 25
sym.discriminant(quadratic)

\[\displaystyle -144\]

This is negative so we know that the equation does not have any real solutions and hence the graph does not cross the x-axis. Since the coefficient of \(x^2\) is positive it means that the graph is above the \(y=0\) line.

By completing the square, show that the minimum point of \(f(x)\) is \(\left(-2, 9\right)\)

a, b, c = sym.Symbol("a"), sym.Symbol("b"), sym.Symbol("c")
completed_square = a * (x - b) ** 2 + c
sym.expand(completed_square)

\[\displaystyle a b^{2} - 2 a b x + a x^{2} + c\]

This gives \(a=4\).

completed_square = completed_square.subs({a: 4})
sym.expand(completed_square)

\[\displaystyle 4 b^{2} - 8 b x + c + 4 x^{2}\]

Comparing the coefficients of \(x\) we have the equation:

\[ - 8 b = 16 \]

equation = sym.Eq(-8 * b, 16)
sym.solveset(equation, b)

\[\displaystyle \left\{-2\right\}\]

Substituting:

completed_square = completed_square.subs({b: -2})
sym.expand(completed_square)

\[\displaystyle c + 4 x^{2} + 16 x + 16\]

Comparing the coefficients of \(x^0\) this gives:

\[c+16=25\]

equation = sym.Eq(c + 16, 25)
sym.solveset(equation, c)

\[\displaystyle \left\{9\right\}\]

completed_square = completed_square.subs({c: 9})
completed_square

\[\displaystyle 4 \left(x + 2\right)^{2} + 9\]

The lowest value of \(f(x)\) is for \(x=-2\) which gives: \(f(-2)=9\) as expected.

Question 5

5. Consider the quadratic: \(f(x)=-3x ^ 2 + 24x - 97\):

Calculate the discriminant of the quadratic equation \(-3x ^ 2 + 24x - 97 = 0\). What does this tell us about the solutions to the equation? What does this tell us about the graph of \(f(x)\)?

quadratic = -3 * x ** 2 + 24 * x - 97
sym.discriminant(quadratic)

\[\displaystyle -588\]

This is negative so we know that the equation does not have any real solutions and hence the graph does not cross the x-axis. Since the coefficient of \(x^2\) is negative it means that the graph is below the \(y=0\) line.

By completing the square, show that the maximum point of \(f(x)\) is \(\left(4, -49\right)\)

a, b, c = sym.Symbol("a"), sym.Symbol("b"), sym.Symbol("c")
completed_square = a * (x - b) ** 2 + c
sym.expand(completed_square)

\[\displaystyle a b^{2} - 2 a b x + a x^{2} + c\]

This gives \(a=-3\).

completed_square = completed_square.subs({a: -3})
sym.expand(completed_square)

\[\displaystyle - 3 b^{2} + 6 b x + c - 3 x^{2}\]

Comparing the coefficients of \(x\) we have the equation:

\[ 6 b = 24 \]

equation = sym.Eq(6 * b, 24)
sym.solveset(equation, b)

\[\displaystyle \left\{4\right\}\]

Substituting:

completed_square = completed_square.subs({b: 4})
sym.expand(completed_square)

\[\displaystyle c - 3 x^{2} + 24 x - 48\]

Comparing the coefficients of \(x^0\) this gives:

\[c-48=-97\]

equation = sym.Eq(c - 48, -97)
sym.solveset(equation, c)

\[\displaystyle \left\{-49\right\}\]

completed_square = completed_square.subs({c: -49})
completed_square

\[\displaystyle - 3 \left(x - 4\right)^{2} - 49\]

The highest value of \(f(x)\) is for \(x=4\) which gives: \(f(4)=-49\) as expected.

Question 6

6. Consider the function \(f(x) = x^ 2 + a x + b\).

Given that \(f(0) = 0\) and \(f(3) = 0\) obtain the values of \(a\) and \(b\).

Substituting 0 in to \(f\) gives:

expression = x ** 2 + a * x + b
expression.subs({x: 0})

\[\displaystyle b\]

This implies that \(b=0\). Substituting back in to the expression:

expression = expression.subs({b: 0})
expression

\[\displaystyle a x + x^{2}\]

Substituting \(x=3\) in to this expression gives:

expression.subs({x: 3})

\[\displaystyle 3 a + 9\]

This gives the equation:

\[ 3 a + 9 = 0 \]

sym.solveset(expression.subs({x: 3}), a)

\[\displaystyle \left\{-3\right\}\]

Our expression is thus:

expression = expression.subs({a: -3})
expression

\[\displaystyle x^{2} - 3 x\]

By completing the square confirm that graph of \(f(x)\) has a line of symmetry at \(x=\frac{3}{2}\)

completed_square = a * (x - b) ** 2 + c
sym.expand(completed_square)

\[\displaystyle a b^{2} - 2 a b x + a x^{2} + c\]

We see that \(a=1\) and. Substituting:

completed_square = completed_square.subs({a: 1})
sym.expand(completed_square)

\[\displaystyle b^{2} - 2 b x + c + x^{2}\]

This gives:

\[ -2b=-3 \]

equation = sym.Eq(-2 * b, -3)
sym.solveset(equation, b)

\[\displaystyle \left\{\frac{3}{2}\right\}\]

Substituting:

completed_square = completed_square.subs({b: sym.S(3) / 2})
sym.expand(completed_square)

\[\displaystyle c + x^{2} - 3 x + \frac{9}{4}\]

Which gives:

\[ c + 9 / 4 = 0 \]

equation = sym.Eq(c + sym.S(9) / 4, 0)
sym.solveset(equation, c)

\[\displaystyle \left\{- \frac{9}{4}\right\}\]

Substituting:

completed_square = completed_square.subs({c: -sym.S(9) / 4})
completed_square

\[\displaystyle \left(x - \frac{3}{2}\right)^{2} - \frac{9}{4}\]

Thus \(x=3/2\) is a line of symmetry.