Proof

• How do you prove that a quadratic is positive for all values of x?

  complete the square and show min point is positive.
• How do you prove a quadratic > x + a ?

  move all to one side and complete square for minimum value of x. sub value into original interval to show it is always larger.
• prove there are no integers where 6x + 9y = 1?

  factorise, 3(2x + 3y) = 1 2x + 3y = ⅓ 2x and 3y are integers. integer + integer can’t equal a fraction.
• Prove root 3 is irrational?

  assume is rational, root 3 =a/b
• Prove there is an infinite number of primes?

  assume finite number of primes, let x = product of all primes (p1xp2xp3xpn) y = x+1. y has no prime factors p1 to pn so either has a prime not listed or is a prime itself
• Prove that sum of rational and irrational number is irrational?

  let rational = a/b let irrational = c (where can’t be expressed as a/b) and let sum = d/e. a/b + C = d/e. Rearrange. c = bd - ae / be. Written as fraction so contradicting.
• X^2 - y^2 = 1 is an example of?

  difference of two squares