Algebraic Proof Flashcards

All 9 cards in this deck

What are the steps to prove a statement about integers is always a multiple of a given number?

e.g. prove the sum of two consecutive odd numbers is always a multiple of $4$

1. Write the integers algebraically: $2n+1$ and $2n+3$
2. Form and simplify the expression: $(2n+1)+(2n+3)=4n+4$
3. Factorise and conclude: $4(n+1)$, so it is always a multiple of $4$

True or false? To prove that $(x+2)(x+5)\equiv x^2+7x+10$ it is enough to check that both sides give the same value when $x=1$.

False. An identity must be true for every value of $x$, so you expand and compare coefficients of like terms; testing values is not a proof.

What are the steps to prove algebraically that a recurring decimal equals a given fraction?

e.g. prove $0.\dot{2}\dot{7}=\frac{3}{11}$

1. Let $x$ be the decimal: $x=0.\dot{2}\dot{7}$
2. Multiply by a power of $10$ so the recurring parts line up: $100x=27.\dot{2}\dot{7}$
3. Subtract and solve: $99x=27$, so $x=\frac{27}{99}=\frac{3}{11}$

What must you show to prove algebraically that a straight line is a tangent to a circle?

e.g. $x-2y=10$ and $x^2+y^2=20$

Substitute the line into the circle equation and show the resulting quadratic has one repeated root, e.g. $(y+4)^2=0$, so there is only one point of intersection.

What are the steps to prove that a quadratic expression is always positive?

e.g. $x^2+6x+11$

1. Complete the square: $x^2+6x+11=(x+3)^2+2$
2. State $(x+3)^2\ge 0$ for all $x$
3. So the expression is at least $2$, hence always positive

True or false? Showing that an expression simplifies to $(n+1)^2$, where $n$ is an integer, proves the expression is always a square number.

True. $(n+1)^2$ is the square of an integer for every integer $n$.

True or false? $x^2$ is always positive, whatever the value of $x$.

False. $x^2\ge 0$; it equals $0$ when $x=0$, so it is never negative but not always positive.

What are the steps to prove algebraically that a straight line is a tangent to a circle?

e.g. $y=x+4$ and $x^2+y^2=8$

1. Substitute the line into the circle: $x^2+(x+4)^2=8$
2. Simplify to a quadratic: $2x^2+8x+8=0$, i.e. $x^2+4x+4=0$
3. Solve: $(x+2)^2=0$, a repeated root, so only one point of intersection — the line is a tangent

True or false? If substituting the equation of a line into the equation of a circle gives a quadratic with two different solutions, the line is a tangent to the circle.

False. Two different solutions mean two points of intersection; a tangent gives a repeated root (one point).