MAST4001: Explain two different methods to solve a quadratic equation and what is meant by a real root: Algebraic Methods Assignment, UOK, UK

Questions

a) Explain two different methods to solve a quadratic equation and what is meant by a real root. For each, describe an instance where it would be appropriate to use this method and give an example calculation.

b) Define the discriminant (sometimes known as determinant) of a quadratic equation and, using your own words, explain how it can be used to determine the number of solutions to the equation. Show example calculations to where one, two, or no real roots are found. Use hand-drawn or software-generated graphs to show the roots of the quadratic and indicate which roots are real.

c) Explain in your own words how to determine whether a geometric series will:

• converge
• diverge
• oscillate Define any algebraic variables you use.

d) Give an example of each type of series and carry out an example calculation to show that it converges, diverges, or oscillates.

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a) Express x² + 4x – 7 in the form (x + p)² – q, where p and q are integers.

b) Hence, or otherwise, find the coordinates of the minimum point of the curve y = x² + 4x – 7.

3. The quadratic equation x² + (3k + 1)x + (4 – 9k), where k is constant, has repeated roots.
a) Show that 9k² + 42k – 15 = 0.

b) Hence find the possible values of k.

4.
a) Find the binomial expansion of (2 + 3x)5, simplifying the terms.

b) Hence find the binomial expansion of (2 + 3x)5 – (2 – 3x)5

5.
a) Evaluate and simplify the following logarithm to find x 2logb 5 + ½ log 9 − log 3 = logo x

c) The formula for the amount of energy E (in joules) released by an earthquake is E = (1.74 × 1019 × 101.44M) where M is the magnitude of the earthquake on the Richter scale.

i. The Newcastle earthquake in 1989 had a magnitude of 5 on the Richter scale. How many joules were released?

ii. In an earthquake in San Francisco in the 1900s the amount of energy released was double that of the Newcastle earthquake. What was its Richter magnitude?

6.

The first term of an infinite geometric series is 96. The common ratio of the series is 0.4.
a) Find the third term of the series.

b) Find the sum to infinity of the series.

7.

An arithmetic series has first term a and common difference d. The sum of the first ten terms of the series is 460.
a) Show that 2a + 9d = 92.

b) Given also that the 25th term of the sequence is 241, find the value of d.

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