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# A proof sorter activity, based on
# https://undergroundmathematics.org/quadratics/proving-quad-formula
format: 1
type: cardsort
resource: E1_RT5_1
title: $\sqrt{2}$ is irrational
note: Sort the cards into order to prove the quadratic formula.
# puzzle size
rows: 5
columns: 1
# Do we shuffle the order of the cards from the order given when
# producing the problem? Say "true" for a proof sorting activity.
# It probably does not make sense to say "false" here but still to
# produce a solution.
shuffleCards: true
# Does this puzzle require production of a solution? If so, the cards
# must be given in the correct order initially.
produceSolution: true
# Default text size on cards
textSize: 5
# Card title
cardTitle: Quadratic formula proof
cardTitleSize: 1
cards:
- Consider $ax^2 + bx + c = 0$, where $a \neq 0$.
- Since $a \neq 0$, we can divide by $a$ to get
$$x^2 + \frac{b}{a} x + \frac{c}{a} = 0.$$
- We complete the square.
- This shows that the original equation is equivalent to
$$\left(x + \frac{b}{2a} \right)^2 - \frac{b^2}{4a^2} + \frac{c}{a} = 0.$$
- Since $x$ appears only once in the equation, we can rearrange this
to solve for $x$.
- 'Get the squared term on one side of the equation:
$$\left(x + \frac{b}{2a} \right)^2 = \frac{b^2}{4a^2} - \frac{c}{a}.$$'
- 'We can rewrite the right-hand side by putting it over a common denominator:
$$\left(x + \frac{b}{2a} \right)^2 = \frac{b^2 - 4ac}{4a^2}.$$'
- We can take the square root of both sides.
- Taking account of the possibility of positive and negative square roots, we see
$$x + \frac{b}{2a} = \frac{\pm\sqrt{b^2 - 4ac}}{2a}.$$
- Subtracting $\frac{b}{2a}$ from both sides and putting the right-hand side over a common denominator gives
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.$$
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