Enrichment, Extension, and Application

Question 1

1 point

Print

Exercises

  1. A quadratic function is of the form \( f(x) = x^2 + bx + c \). The roots of the equation \( f(x) = 0 \) are \( 1 \) and \( k \). If \( f(2) = 5 \), determine \( k \).
  2. The parabola \( y = x^2 - 2x + 4 \) is translated \( p \) units to the right and \( q \) units down. The \( x \) - intercepts of the resulting parabola are \( 3 \) and \( 5 \). Determine the values of \( p \) and \( q \).
    1. For the quadratic equation \( ax^2 + bx + c = 0 \), where \( a \neq 0 \), show that the sum of the roots is \( - \dfrac{b}{a} \) and the product of the roots is \( \dfrac{c}{a} \).
    2. The roots of \( x^2 + cx + d = 0 \) are \( a \) and \( b \). The roots of \( x^2 + ax + b = 0 \) are \( c \) and \( d \). If \( a\), \(b\), \(c \), and \( d \) are all non-zero, determine the value of \( a + b + c + d \).
  3. Prove that the line with equation \( y = 2x - 1 \) does not intersect the curve with equation \( y = x^4 + 3x^2 + 2x \).
    1. If \( r_1\), \(r_2 \), and \( r_3 \) are the roots of the cubic equation \( ax^3 + bx^2 + cx + d = 0 \), find equations which relate \( r_1\), \(r_2 \), and \( r_3 \) to the values of the coefficients, \( a\), \(b\), \(c \), and \( d \).
    2. If \( a \), \( b \), and \( c \) are the roots of the equation \( x^3 - 3x^2 + mx + 24 = 0 \) and \( - a \) and \( - b \) are the roots of the equation \( x^2 + nx - 6 = 0 \), find the value of \( n \).
  4. A quadratic equation \( ax^2 + bx + c = 0 \), where \( a\), \(b \), and \( c \) are not zero has real roots. Prove that \( a\), \( b \) and \( c \) cannot be consecutive terms of a geometric sequence. (Note: A geometric sequence is a sequence in which each term after the first is obtained from the previous term by multiplying it by a non-zero constant. For example, \( 3, 6, 12, 24, \ldots \) is a geometric sequence).
  5. The function, \( \text{max}\left[ a,b \right] \), is defined to be the larger of \( a \) or \( b \) (if \( a = b \), then \( \text{max}\left[a, b \right] = a = b \)). If \( x \) is a real number, and \begin{align*} g(x) & = - x^3 + 11x^2 - 24x\\ h(x) & = - x^2 + 6x + 8 \\ f(x) & = \text{max}[g(x),h(x)] & (0 \le x \le 7) \end{align*} determine the values of \( x \) for which \( f(x) = h(x) \), \( 0 \le x \le 7 \).
  6. The equations \( x^2 + 5x + 6 = 0 \) and \( x^2 + 5x - 6 = 0 \) each have integer solutions. However, only one of the equations in the pair \( x^2 + 4x + 5 = 0 \) and \( x^2 + 4x - 5 = 0 \) has integer solutions.
    1. Show that if \( x^2 + px + q = 0 \) and \( x^2 + px - q = 0 \) both have integer solutions, then it is possible to find positive integers \( a \) and \( b \) such that \( p^2 = a^2 + b^2 \) (i.e., \( a,b,p \) is a Pythagorean triple).
    2. Determine \( q \) in terms of \( a \) and \( b \).
  7. Identify the degree of the polynomial and find the value of \( f(x) \) that satisfies the following:
    1. \( f(x) - f(x - 1) = 4 \) with \( f(0) = 4 \)
    2. \( f(x) - 2f(x - 1) + f(x - 2) = 6 \) with \( f(1) = 6, f(0) = 1 \)
  8. If \( x \) and \( y \) are real numbers,
    1. Prove that \( x^4 + 4y^4 + 1 \ge 4xy \).
    2. Determine all values of \( x \) and \( y \) for which \( x^4 + 4y^4 + 1 = 4xy \).
© CEMC and University of Waterloo, Powered by Maplesoft