Notes
Four solving methods
- Factoring: set one side to 0, factor, and use the zero product property.
- Square roots: isolate the squared part and take $\pm$ the root. $(x - 2)^2 = 25 \Rightarrow x = 7$ or $-3$.
- Completing the square: add $\displaystyle \left(\frac{b}{2}\right)^2$ to both sides of $x^2 + bx = c$, then use square roots.
- Quadratic formula: $\displaystyle x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ works for every quadratic.
The discriminant
- $b^2 - 4ac > 0$: two real solutions. $= 0$: one real solution. $< 0$: no real solutions (two complex).
Complex numbers
- $\displaystyle i = \sqrt{-1}$ and $i^2 = -1$. Complex numbers have the form $a + bi$.
- Add real and imaginary parts separately. Multiply like binomials, then replace $i^2$ with $-1$.
- The conjugates $a + bi$ and $a - bi$ multiply to the real number $a^2 + b^2$.
Topics in this unit
Each topic has its own short notes and practice.
Practice
$2x^2 = 18$, so $x^2 = 9$ and $x = \pm 3$.
$(x + 5)(x - 1) = 0$.
$x^2 - 6x = -2$. Add 9: $(x - 3)^2 = 7$.
The discriminant is $1 + 24 = 25$, so $\displaystyle x = \frac{-1 \pm 5}{6}$. That gives $\displaystyle \frac{4}{6} = \frac{2}{3}$ and $-1$.
$b^2 - 4ac = 9 - 32 = -23$, which is negative.
$4 - i^2 = 4 - (-1) = 5$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
A rectangle is 3 meters longer than it is wide. Its area is 40 square meters. Let $w$ be the width.
(a) Write a quadratic equation in standard form for the area.
Show the answer and scoring guide
$w(w + 3) = 40$, so $w^2 + 3w - 40 = 0$.
Check off the points you earned:
(b) Solve by factoring and state the dimensions.
Show the answer and scoring guide
$(w + 8)(w - 5) = 0$, so $w = 5$ (reject $w = -8$). The width is 5 m and the length is 8 m.
Check off the points you earned:
(c) If the area were 50 square meters, use the quadratic formula to find the width to the nearest hundredth.
Show the answer and scoring guide
$w^2 + 3w - 50 = 0$ gives $\displaystyle w = \frac{-3 + \sqrt{9 + 200}}{2} = \frac{-3 + \sqrt{209}}{2} \approx 5.73$ meters. The negative root is rejected.
Check off the points you earned: