**MA310 Homework #1
Due Monday, January 25
**

**
**

**Let's explore what happens when the leading coefficient in a quadratic equation approaches zero. Consider the function .**- 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? - Algebraic analysis: As
*a*approaches zero, the formula for*f*(*x*) approaches*g*(*x*)=5*x*-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. - 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.

- Graphical exploration: On the same coordinate
system draw the graph of this function for several values of
**In this problem we try to find a formula for the square root of a general complex number.**- Find all square roots of
*i*. - 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 . - 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?)
- Prove that one square root of is . What is the other square root? How can you find you find them graphically?
- 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?) - 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.

- Find all square roots of

Wed Jan 27 08:17:21 EST 1999