9 of 9 Items .... Source: David Butler

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What do you think of this method of completing the square?
Prof Butler likes it because you can go direct from factored form to vertex form.






\( x^2 + 12x + 35 \)
\( = x(x + 12) + 35 \)
\( = (x + 6 - 6)(x + 6 + 6) + 35 \)
\( = (x + 6)^2 - 36 + 35 \)
\( = (x + 6)^2 - 1 \)

How does the special factoring of \(a^2 - b^2 \) come into all of this?

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Some triangles have the same perimeter in \(cm\) as area in \(cm^2\), like the 6cm-8cm-10cm triangle.






Can you show that if a triangle DOES have its perimeter in \(cm\) equal to its area in \(cm^2\), then all the edges must be longer than 4cm?

Prof. Butler's explanation is found at the twitter link in the source.

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Consider the fractions \( \dfrac{5}{29} \) and \( \dfrac{1}{2} \).

Adding tops and bottoms produces a number in between, so \( \dfrac{6}{31}\) is in between \( \dfrac{5}{29} \) and \( \dfrac{1}{2} \).

But \(\dfrac{6}{31} \) is less than half, so it must be \( \dfrac{5}{29} < \dfrac{6}{31} < \dfrac{1}{2} \).

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Use only the five numbers 10, 100, 1000, 10 000, 100 000,
with each appearing exactly once, and as many of +, -, ×, ÷ (and brackets) as you like, to make numbers with no zeros in their digits.

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Consider these expressions:

\( 7+8 \)

\( 8+7 \)

\( 7-8 \)

\( 8-7 \)

\( 7*8 \)

\( 8*7 \)

\( 7÷8 \)

\( 8÷7 \)

If you were to replace the \( 8 \)'s with \( 3+5 \), in which expressions would you need brackets around the \( 3+5 \)?

What if you replace \( 8 \) with \( 2*4 \), or with \( 2^3 \)?

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What do you think about this description?
Is this efficient and useful or misleadingly simplistic?

What the leading coefficient of a polynomial does to a graph:

Positive Leading coefficient => right-hand end points up
Negative Leading coefficient => right-hand end points down

Degree odd => ends point opposite ways
Degree even => ends point same way

.: [ALG2], [David Butler], [New Understanding].

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A polynomial with integer coefficients, as a function, produces an integer output for every integer input.

I wonder: if a polynomial as a function *does* produce an integer output for every integer input, *must* its coefficients be integers?

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To which David Butler added:

May I please add one thing, for the purposes of helping your student with future problems?
It might be very important to consider what happens for similar equations, such as THESE two:

\(e^{2x} + 9e^x + 18 = 0\)

\(e^{2x} + 9e^x - 10 = 0\)

.: [PRE-CALC], [David Butler], [Notice, Wonder].

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Trace the edge of the Serpinski Figure below with one continuous line, without going over any line twice.