MA310 Homework #1
Due Monday, January 25

1. Let's explore what happens when the leading coefficient in a quadratic equation approaches zero. Consider the function .
1. Graphical exploration: On the same coordinate system draw the graph of this function for several values of a between -4 and 4. What is happening to the graph as a approaches zero? What is happening to the x-intercepts?
2. Algebraic analysis: As a approaches zero, the formula for f(x) approaches g(x)=5x-3. What is the x-intercept for g(x)? Prove algebraically that one of the x-intercepts for f(x) (i.e., one of the solutions for the quadratic equation f(x)=0) approaches the x-intercept for g(x) as a approaches zero.
3. General algebraic analysis: Given a general quadratic equation where , prove that one of the solutions approaches the solution to bx+c=0 as a approaches zero. For convenience, assume that b>0.
2. In this problem we try to find a formula for the square root of a general complex number.
1. Find all square roots of i.
2. Every complex number a+bi can be represented as a point P=(a,b) in the plane. Assume . Let and let be the angle from the positive x-axis to the ray OP, where O is the origin. Explain why .
3. Suppose you want to multiply by . Prove that the product is . (I.e., to multiply two complex numbers, multiply their ``lengths'' and add their ``angles..'') (Hint: Do you remember your trigonometric angle addition formulas?)
4. Prove that one square root of is . What is the other square root? How can you find you find them graphically?
5. Find all square roots of 1+i. Express your final answer without using sines and cosines. (Hint: Do you remember your trigonometric half-angle formulas?)
6. Find a general formula for the square roots of a+bi in terms of a and b. For convenience, assume that and . Suggestion: Use the half-angle formulas and the fact that and . Confirm your answer by squaring it.