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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 \( 24=\frac{1}{3}(x-1)^2-3\)\(x = 10 \quad x = -8\) \( x^2+5x+5 = 26-x^2-6x \)\(x = 3/2 \quad x = -7\) \( 0=x^2+11x+30\)\(x = -5 \quad x = -6\) -
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.
\[P(t) = -\frac{t^{2}}{2000}+\frac{t}{9}+\frac{21}{5}\]
- 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?
- What does the model indicate the population was in 1980?
- 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.
- 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
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Consider the parabola and line defined by these equations,
where \(\bm{p}\) is some undetermined parameter.
\(y = x^2 - 16x + 55 \)\(y = \bm{p}-2x \)
- What is a formula, in terms of \(\bm{p},\) for the \(x\)-coordinate(s) of the point(s) where this parabola and line intersect?
- For what value of \(\bm{p}\) will the parabola and line intersect at exactly one point?