**15. P2:** a) Show that the solutions of the equation (x^2 – 6x – 43 = 0) can

**15. P2:**
a) Show that the solutions of the equation (x^2 – 6x – 43 = 0) can be written in the form (x = p pm frac{qsqrt{13}}{13}), where (p) and (q) are positive integers.
(4 points)
b) Based on the above or any other method, solve the inequality (x^2 – 6x – 43 le 0).
(2 points)
**16. P1:**
A function (f) is defined by (f(x) = 3(x-1)^2 – 18), (x in mathbb{R}).
a) Write (f(x)) in the form (ax^2 + bx + c), where (a), (b), and (c) are constants.
(2 points)
b) Find the coordinates of the vertex of the graph of (f).
(1 point)
c) Find the equation of the axis of symmetry of the graph of (f).
(1 point)
d) Indicate the range of (f).
(2 points)
e) The graph of (f) is translated by the vector (begin{pmatrix} 2 \ -1 end{pmatrix}) to form a new curve that represents a new function (g(x)).
Find (g(x)) in the form (px^2 + qx + r), where (p), (q), and (r) are constants.
(3 points)
**17. P1:**
a) Solve the equation (8x^2 + 6x – 5 = 0) by factorization.
(2 points)
b) Determine the range of values of (k) for which the equation (8x^2 + 6x – 5 = k) has no real solutions.
(3 points)
**18. P1:**
Consider the function (f(x) = -x^2 – 10x + 27), (x in mathbb{R}).
a) Show that the function (f) can be expressed in the form (f(x) = a(x-h)^2 + k), where (a), (h), and (k) are constants.
(3 points)
b) Based on the above, write the coordinates of the vertex of the graph of (y = f(x)).
(1 point)
c) Based on the above, write the equation of the axis of symmetry of the graph of (y = f(x)).
(1 point)
**19. P1:**
The quadratic curve (y = x^2 + bx + c) cuts the x-axis at ( (10, 0) ) and has the equation of the line of symmetry (x = frac{5}{2}).
a) Find the values of (b) and (c).
(4 points)
b) Based on the above, or any other method, find the other two coordinates where the curve intersects the y-axis.
(2 points)
**20. P1:**
Consider the function (f(x) = 2x^2 – 4x – 8), (x in mathbb{R}).
a) Show that the function (f) can be expressed in the form (f(x) = a(x-h)^2 + k), where (a), (h), and (k) are constants.
(3 points)
b) The function (f(x)) can be obtained through a sequence of transformations of (g(x) = x^2). Describe each transformation in order.
(3 points)
**21. P1:**
Consider the equation (f(x) = 2kx^2 + 6x + k), (x in mathbb{R}).
a) For the case where the equation (f(x) = 0) has two equal real roots, find the possible values of (k).
(4 points)
b) For the case where the equation of the axis of symmetry of the curve (y = f(x)) is (x + 1 = 0), find the value of (k).
(2 points)
c) Solve the equation (f(x) = 0) when (k = 2).
(3 points)
**22. P1:**
A curve (y = f(x)) passes through the points with coordinates (A(-12, 10)), (B(0, -16)), (C(2, 9)), and (D(14, -10)).
a) Write the coordinates of each point after the curve has been transformed by (f(x) rightarrow f(2x)).
(4 points)
b) Write the coordinates of each point after the curve has been transformed by (f(x) rightarrow f(-x) + 3).
The homework is in Spanish I’m going to download the picture for you but this is the translation.