Solving Quadratic Equations

  1. Figure out which value(s) of \(x\) satisfy each of the following quadratic equations. Be sure to express those values as concisely as possible.

    \( (2x-3)(x+1) = 0\)
    \(x = 3/2 \quad x = -1\)
    \( x^2+x+1=3(x+3)\)
    \(x = -2 \quad x = 4\)
    \( x^2-99=45\)
    \(x = \pm 12\)
    \( x^2+6x+6=5x+13\)
    \(x = \frac{1}{2}\Bigl(-1 \pm \sqrt{29}\Bigr)\)
    \( 3x=17+x^2\)
    no solution
    \( 24=\frac{1}{3}(x-1)^2-3\)
    \(x = 10 \quad x = -8\)
    \( x^2+5x+5 = 26-x^2-6x \)
    \(x = 3/2 \quad x = -7\)
    \( 0=x^2+11x+30\)
    \(x = -5 \quad x = -6\)
  2. Suppose that, based on historical data from the UN, the population of the earth, in billions, can be modelled by this quadratic function \(P\) measured \(t\) years since 1980. \[P(t) = -\frac{t^{2}}{2000}+\frac{t}{9}+\frac{21}{5}\]
    1. According to this model, the population of the earth will hit a maximum and then begin to decline. What is the range of years during which the population will be declining?
    2. What does the model indicate the population was in 1980?
    3. Since the population was less than 5 billion people in 1980, but is more than 5 billion people today, there must be a time between now and then at which the earth’s population was exactly 5 billion people. Find the year during which the model indicates this happened. Then since the model indicates the earth’s population will eventually decline to zero, it must become exactly 5 billion again in the future; calculate the year this happens too.
  3. Given a quadratic equation in standard form \(ax^2+bx+c\,\) how can you quickly tell whether or not it has real roots? Hint: look at the quadratic formula and figure out what “goes wrong” algebraically when there are no real roots.

Challenges

  1. Consider the parabola and line defined by these equations, where \(\bm{p}\) is some undetermined parameter.
    \(y = x^2 - 16x + 55 \)
    \(y = \bm{p}-2x \)
    1. What is a formula, in terms of \(\bm{p},\) for the \(x\)-coordinate(s) of the point(s) where this parabola and line intersect?
    2. For what value of \(\bm{p}\) will the parabola and line intersect at exactly one point?