Exercises

1. Simplify the following expressions:

   1. \(\frac{3}{\sqrt{3}}\)

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

   3. \(8^0\)

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

2. Solve the following equations:

   1. \(x + 3 = -1\)

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

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

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

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

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

   2. 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.

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

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

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

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

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

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

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

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

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