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David Williams @bewilda.bsky.social · 06/10/2026

The textbook from my final year of secondary school (1971). Ax² + Bxy + Cy² = K then tan(2α) = B/(A - C) and the double angle formula results in tan(α). A' + C' = A + C A' - C' = Bcosec(2α) Solve for A' and C' A'x² + C'y² = K

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David Williams @bewilda.bsky.social · 06/10/2026

Rotating the axes through arctan(2) transforms the equation to x²/2 + y² = 1 ∴ r = 1 and area of the ellipse = π x √2 x 1 = π√2 Shaded area = π(√2 - 1) As an aside: For any ellipse in the form Ax² + Bxy + Cy² = K, the enclosed area is given by: Area = 2πK/√(4AC - B²)

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David Williams @bewilda.bsky.social · 30/09/2026

Common tangent (x - 2y)/(2x + 2y) = -x/y 2x² + 3xy - 2y² = 0 (2x - y)(x + 2y) =0 x = y/2 discard as subs. into hyperbola leads to no real solutions x = -2y Two solutions (-2,1) and (2,-1) r² = 5 Area = 5π

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David Williams @bewilda.bsky.social · 21/09/2026

Multiply both sides by 1 + 2i Divide both sides by 5 Multiply both sides by 4 Complete the square [2z + (1 - 2i)]² = 25 z = -3 + i or 2 + i

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David Williams @bewilda.bsky.social · 19/09/2026

Another quadratic in disguise. x⁴ + 6x³ + 7x² - 6x - 8 {Let u = x² + 3x}

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David Williams @bewilda.bsky.social · 06/09/2026

To make it clear from the outset that you're GOM. [God of Mathematics or is it Grumpy Old Man?]

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David Williams @bewilda.bsky.social · 03/09/2026

Equation of tangent: y = mx + 3 - 2m x² = mx + 3 - 2m 4x² - 4mx = -8m + 12 (2x - m)² = m² - 8m +12 =(m - 2)(m - 6) ∴ m = 2 or 6 [Point of contact (m/2, m²/4)] y = 2x - 1 or y = 6x - 9 www.desmos.com/calculator/e...

desmos.com

Tangent from a point

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David Williams @bewilda.bsky.social · 03/09/2026

Fun facts: Tangents to y = ax² + bx + c, which pass through the origin, meet the parabola at x = ±√(c/a). The product of the slopes of the tangents which pass through the origin is b² - 4ac.

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David Williams @bewilda.bsky.social · 06/07/2026

The referee was outstanding in such a pressure cooker environment.

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David Williams @bewilda.bsky.social · 15/05/2026

Let y = 3^x y² -(4a + 5)y +4a² + 10a = 0 [y - (2a + 5)][y - 2a] = 0 3^x = 2a + 5 or 2a No real roots → 2a + 5 ≤ 0 a ≤ -5/2 Answer: -5/2

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David Williams @bewilda.bsky.social · 03/02/2026

Equation of circle: (x - r)² + (y - r)² = r² Curve and circle both symmetric to y = x. Point of contact (xₒ, xₒ). ∴ xₒ = ¼ Substitute to find r.

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David Williams @bewilda.bsky.social · 26/01/2026

Australia Day (26/1). In my neck of the woods the temperature reached 45⁰ C. It's 5:54 am (27/1) and it's still 36⁰ C. (p, (188 - 3p)/7) lies on the line. y - x > 0 188 - 10p > 0 p < 18.8 p = 16 yields the first integer solution → (16, 20)

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David Williams @bewilda.bsky.social · 25/01/2026

Just an observation: If p is of the form 10a + 3 then p² - p - 1 is divisible by 5. p = 3 → 5 (divisible by 5 but prime) p = 13 → 155 p = 23 → 505 p = 43 → 1805

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David Williams @bewilda.bsky.social · 21/01/2026

127 is the first Friedman prime. My grandson and I were watching Australian Open tennis singles matches today and he asked how many players are in the main draw. My response was 128 and there has to be 127 matches to decide the winner (127 players have to lose). Tennis is so boring.

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David Williams @bewilda.bsky.social · 28/11/2025

1981 = 13x13x13 - 6x6x6 (unlucky and evil at the same time). With apologies to one of my daughters who was born on 13/6/1981.

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David Williams @bewilda.bsky.social · 27/11/2025

It's an even function. X-intercepts: x = kπ, k ≠ 0. Stationary points where tanx = x. y ≤ |1/(x - sinx)| x → ±∞, y → 0

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David Williams @bewilda.bsky.social · 27/11/2025

Multiply numerator and denominator by x + √(1 + x²) followed by some algebraic manipulation and then the u substitution. This was the method I was shown in the dark ages.

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David Williams @bewilda.bsky.social · 09/10/2025

I do this, too. Usually preceded by Descartes' rule of signs. In South Australia, students only have examinations in their final year of school and are allowed to use graphics calculators. In the majority of the other states CAS calculators are allowed.

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David Williams @bewilda.bsky.social · 05/10/2025

Just for fun: Disguised Quadratics. Solve x⁴ + 6x³ + 4x² - 15x - 50 = 0. [Hint: Let u = x² + 3x]

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David Williams @bewilda.bsky.social · 27/09/2025

Just for fun when you know the answer.

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David Williams @bewilda.bsky.social · 05/07/2025

The parabola and the straight line y = mx + p intersect where ax² + bx +c = mx + p ax² + (b - m)x + c - p = 0 If the line is a tangent then (b - m)² - 4a(c - p) = 0 m = b ± 2√(a(c - p)) ∴ m₁m₂ = b² - 4a(c - p) If the tangents pass through the origin then p = 0 and m₁m₂ = b² - 4ac (= ▵)

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David Williams @bewilda.bsky.social · 03/07/2025

I'm intrigued. The abbreviation cisθ is used in exams in every state in Australia.

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David Williams @bewilda.bsky.social · 26/06/2025

Perhaps it's a defence mechanism (they have to make sense of the idiocy of ACARA for 40 weeks of the year) or they've heard once too often "I wish Eddie was my teacher."

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David Williams @bewilda.bsky.social · 16/06/2025

Find the positive integer A such that exactly two of the following statements are true: (a) A + 82 is the square of an integer; (b) the last digit of A is 5; (c) A − 7 is the square of an integer.

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David Williams @bewilda.bsky.social · 07/06/2025

Even more, the [3, 7, 8] and [3, 5, 7] triangles always living in the shadow of the[3, 4, 5] triangle.

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David Williams @bewilda.bsky.social · 24/04/2025

Consider the triangle ABC with AB = n^2, BC = 2n + 1 and AC = n(n+ 1) for all n > 1. Use the cosine rule and the sine rule to find angles B and C for at least 3 integer values of n. What do you notice? Can you prove your conjecture? What ”sublime” triangle is formed when AC = BC?

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David Williams @bewilda.bsky.social · 03/04/2025

coshx is an even function. If the solutions are x₁ and x₂ then x₁ + x₂ = 0, x₁ = -x₂

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David Williams @bewilda.bsky.social · 30/03/2025

Is this a valid approach? ∫ln(x - 4)dx = (x - 4)ln(x - 4) - ∫(x - 4)/(x - 4)dx = (x - 4)ln(x - 4) - ∫dx = (x - 4)ln(x - 4) - x + c

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David Williams @bewilda.bsky.social · 07/03/2025

I lean towards method 1 over 2.

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David Williams @bewilda.bsky.social · 07/03/2025

Or repeated synthetic division.

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David Williams @bewilda.bsky.social · 12/02/2025

It's the 43rd day of the year and it's a stifling 43⁰ C today in the city where I live. Fun fact: -40⁰ C = -40⁰ F.

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David Williams @bewilda.bsky.social · 11/02/2025

Fun fact: Today is the 42nd day of the year (Feb. 11th). 10! seconds is exactly 42 days.

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David Williams @bewilda.bsky.social · 06/02/2025

This curve is symmetric about the line y = x. The two branches are inverses. In this case, if the gradient of one tangent is a/b, the gradient of the other is b/a. tanα = |(a² - b²)/2ab|, α is acute.

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David Williams @bewilda.bsky.social · 03/02/2025

It's an isosceles right triangle. Pythagoras Theorem.

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David Williams @bewilda.bsky.social · 02/02/2025

The Australian 50 cent coin. In a more delusional moment, I once made a claim that I could teach an entire middle school geometry course using this coin.

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David Williams @bewilda.bsky.social · 31/01/2025

If your'e Australian, Chopper is definitely number one.

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David Williams @bewilda.bsky.social · 28/01/2025

So y = x is not a function unless domain is stipulated. Desmos and Geogebra are able to draw both functions without a domain being listed. 1 - 8xe^(4x) >= 0 x <= 0.088 (3sf)

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David Williams @bewilda.bsky.social · 28/01/2025

Both the explicit relations I have listed are functions.

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