SecondDegree Equations and Inequalities

1
Second-Degree Equations and Inequalities
2
Learning Objectives

• Quadratic form equations.
• Applications.
• Graphs of second-degree equations.

3
Quadratic Form Equations

• Some equations can be transformed into a
  quadratic equation by substitution. For example,
  if u is a substitution, solve au2 bu c 0, a
  gt0, then substitute u to find the solution.

4
Quadratic Form Equations

• Example
• We check that all these roots satisfy the
  original equation.

5
Quadratic Form Equations

• Example
• We check that all these roots satisfy the
  original equation.

6
Applications

• Example a ladder leans against a building. The
  ladder is 20 ft long. The distance to the top of
  the ladder is 4 ft greater than the distance d
  from the building. Find the distance d.
• The Pythagorean equation gives
• d2 (d4)2 202
• and the following quadratic equation
  2d2 8d 384 0
• the solutions are d-16 and d12.
• d-16 is not a possible solution.
• The solution is d12.

20
d4
d
7
Graphs of Second-degree Equations

• The graph of ax2 bx c 0, a gt0, are the
  intercepts of the parabola y ax2 bx c.

8
Graphs of Second-degree Equations

• Parabola y ax2 bx c can be drawn by
  calculating
• x-intercepts are the solutions of the quadratic
  equations ax2 bx c 0.
• the summit of the parabola is solution of the
  first degree equation 2axb 0 (x-b/2a). It can
  be a maximum or a minimum.
• the concavity of the curve is
• if agt0, the curve opens upward
• if alt0, the curve opens downward.

9
Graphs of Second-degree Equations

• Parabola y -x2 - 4x 1

10
Graphs of Second-degree Equations

• Parabola x 2y2 - y 2
• when the equation is not given in the form
  yf(x), use with(plots) and implicitplot.

11
Quadratic and Rational Inequalities

• A quadratic inequality is of the form ax2 bx
  c lt 0, (or with symbols gt, ?, ?).
• To solve these inequalities
• Write the inequality in standard form (agt0).
• Find the critical values of the quadratic
  equation ax2 bx c 0.
• Construct a sign diagram for the factors in the
  intervals determined by the critical values.
• Identify and record the solution set.

12
Quadratic and Rational Inequalities

• Properties A quadratic function of the form ax2
  bx c is of the sign of a outside of the
  roots, and of the opposite sign of a inside the
  roots.
• Rational inequalities same method starting from
  P(x)/Q(x) lt 0.
• General inequalities write the inequality in
  standard form, then if odd power, inclusion of
  one of the two intervals, and if even power,
  inclusion of both or neither of the intervals.

13
Quadratic and Rational Inequalities
14
Graphing with Maple

• If equation is of the form yf(x), graph with
  plot(f(x), x-a..a, linestylePLAIN)
• If equation is in the form g(x,y)c, plot
  withwith(plots)implicitplot( g(x,y)c,
  x-a..a, y-b..b)

15
Graphing with Maple

• Solving quadratic equations, rationals and
  inequalities with Maple