# Enrichment, Extension, and Application

1 point

## 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 $$x^4 + 4y^4 + 1 = 4xy$$.