Algebra Toolkit Flashcards

All 34 cards in this deck

What are the steps to expand and simplify a product of two brackets?

e.g. $(2x+3)(x-4)$

1. Multiply every term in the first bracket by every term in the second: $2x^2 - 8x + 3x - 12$
2. Collect like terms: $2x^2 - 5x - 12$

What are the steps to expand and simplify a product of three brackets?

e.g. $(x+1)(x+2)(x+3)$

1. Expand any two brackets: $(x+1)(x+2) = x^2+3x+2$
2. Multiply every term of that quadratic by each term of the remaining bracket: $x^3+3x^2+2x+3x^2+9x+6$
3. Collect like terms: $x^3+6x^2+11x+6$

What does "factorise fully" mean?

Take out the highest common factor of all the terms (numbers and letters), so the bracket has no common factor left.
e.g. $6x^2y + 21xy = 3xy(2x+7)$

What happens to the signs when you subtract a bracketed expression?

e.g. $\ldots - (3x^2 - 5x + 2)$

Every term inside the bracket changes sign.
e.g. it becomes $\ldots - 3x^2 + 5x - 2$

True or false? Leaving a term as $2x^3 \times 3x^2$ in an expansion is acceptable in the final answer.

False. Terms must be fully processed — it must be written as $6x^5$.

What are the steps to factorise $x^2 + bx + c$?

e.g. $x^2 + 7x + 12$

1. Find two numbers with product $c$ and sum $b$: product $12$, sum $7$ gives $3$ and $4$
2. Write the brackets with those numbers: $(x+3)(x+4)$

What is the factorised form of $a^2 - b^2$ (the difference of two squares)?

$(a+b)(a-b)$, in either order.
e.g. $x^2 - y^2 = (x+y)(x-y)$

True or false? $x^2 + 9$ factorises as $(x+3)(x-3)$.

False. $(x+3)(x-3) = x^2 - 9$; a sum of two squares does not factorise.

What are the steps to factorise a difference of two squared brackets?

e.g. $(2x+3)^2 - (2x-5)^2$

1. Treat it as $A^2 - B^2 = (A+B)(A-B)$ with $A = 2x+3$, $B = 2x-5$
2. Simplify each bracket: $(4x-2)(8)$
3. Tidy the answer: $16(2x-1)$

What are the steps to factorise $ax^2 + bx + c$ by splitting the middle term?

e.g. $2x^2 + 7x + 3$

1. Find two numbers with product $ac$ and sum $b$: product $6$, sum $7$ gives $6$ and $1$
2. Split the middle term: $2x^2 + 6x + x + 3$
3. Factorise in pairs: $2x(x+3) + 1(x+3) = (x+3)(2x+1)$

What should you always check for before trying to factorise a quadratic into two brackets?

e.g. $2x^2 + 10x + 12$

A common factor of all three terms — take it out first.
e.g. $2(x^2+5x+6) = 2(x+2)(x+3)$

What are the steps to factorise a two-variable quadratic?

e.g. $3x^2 - 5xy - 2y^2$

1. Each bracket has an $x$ term and a $y$ term: $(3x\ \ )(x\ \ )$
2. Find the $y$ terms whose product gives $-2y^2$ and whose cross terms give $-5xy$: $+y$ and $-2y$
3. Answer: $(3x+y)(x-2y)$

True or false? In $x^2 + bx + c$ with $c$ positive and $b$ negative, both numbers in the brackets are negative.

True. Two negatives multiply to a positive $c$ and add to a negative $b$.

How do you check a factorisation is correct?

e.g. you claim $2x^2+7x+3 = (x+3)(2x+1)$

Expand the brackets and check you get back the original expression.
e.g. $(x+3)(2x+1) = 2x^2+x+6x+3 = 2x^2+7x+3$ ✓

What is the completed square form of $x^2 + bx + c$?

$\left(x+\frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2 + c$
e.g. $x^2+6x+1 = (x+3)^2 - 9 + 1 = (x+3)^2 - 8$

What are the steps to write $ax^2+bx+c$ in the form $k(x+m)^2+n$?

e.g. $2x^2 - 16x + 7$

1. Factor out the coefficient of $x^2$ from the first two terms: $2(x^2-8x)+7$
2. Complete the square inside: $2[(x-4)^2-16]+7$
3. Expand and simplify: $2(x-4)^2 - 25$

What does $k(x+m)^2+n$ expand to?

$kx^2 + 2kmx + km^2 + n$

For $k(x+m)^2+n$ with $k>0$, what is the minimum value and where does it occur?

Minimum value $n$, occurring at $x=-m$.
e.g. $2(x-4)^2-25$ has minimum $-25$ at $x=4$

For $k(x+m)^2+n$ with $k<0$, what is the maximum value and where does it occur?

Maximum value $n$, occurring at $x=-m$.
e.g. $-3(x+2)^2+7$ has maximum $7$ at $x=-2$

True or false? The minimum of $(x-3)^2+5$ is $5$ and it occurs at $x=-3$.

False. The minimum is $5$ but it occurs at $x=3$.

What is the quadratic formula for solving $ax^2+bx+c=0$?

$x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$

If a product of two factors equals zero, what must be true?

e.g. $(x+3)(2x-1)=0$

At least one factor is zero.
e.g. $x+3=0$ or $2x-1=0$, so $x=-3$ or $x=\frac{1}{2}$

What are the steps to solve a quadratic equation by factorising?

e.g. $x^2 + x - 6 = 0$

1. Make sure one side is $0$ — it already is
2. Factorise: $(x+3)(x-2)=0$
3. Set each bracket to zero: $x=-3$ or $x=2$

What are the steps to solve a quadratic given in completed square form?

e.g. $(x-1)^2 - 5 = 0$

1. Isolate the squared bracket: $(x-1)^2 = 5$
2. Take the square root of both sides, with $\pm$: $x-1 = \pm\sqrt{5}$
3. Solve for $x$: $x = 1 \pm \sqrt{5}$

What are the steps to solve an equation that becomes a quadratic after expanding?

e.g. $x(x+2)=15$

1. Expand: $x^2+2x=15$
2. Collect all terms on one side: $x^2+2x-15=0$
3. Factorise and solve: $(x+5)(x-3)=0$, so $x=-5$ or $x=3$

True or false? To solve $x^3 - 3x^2 = 0$ you can divide both sides by $x^2$.

False. You lose the solution $x=0$ — factorise to $x^2(x-3)=0$ giving $x=0$ or $x=3$.

What does the symbol $\equiv$ mean in algebra?

It shows an identity — the two sides are equal for every value of the variable.

What does "equating coefficients" in an identity mean?

The coefficients of each power of $x$ (and the constant terms) must be equal on both sides.
e.g. in $ax^2+\ldots \equiv 5x^2+\ldots$, $a=5$

What are the steps to find unknown constants in an identity?

e.g. $x^2+6x+11 \equiv (x+a)^2+b$

1. Expand the side with the unknowns: $x^2+2ax+a^2+b$
2. Equate coefficients of $x$: $2a=6$, so $a=3$
3. Equate constants: $9+b=11$, so $b=2$

Apart from comparing coefficients, what other method finds unknown constants in an identity?

Substitute convenient values of $x$ into both sides.
e.g. putting $x=0$ compares the constant terms

What are the steps to form and solve an equation from a worded context?

e.g. a rectangle of width $x$ and length $x+3$ has area $40$

1. Write an expression for the quantity described: area $= x(x+3)$
2. Set it equal to the given value and rearrange to $=0$: $x^2+3x-40=0$
3. Solve and interpret: $(x+8)(x-5)=0$, so $x=5$ (reject $x=-8$)

When a quadratic from a length, area or volume problem gives a negative solution, what do you do?

Reject it — a length cannot be negative — and state only the valid solution.

What are the steps to find where a line meets a circle centred on the origin?

e.g. $x^2+y^2=25$ and $x = 2y-5$

1. Substitute the line into the circle equation: $(2y-5)^2+y^2=25$
2. Expand and solve the quadratic: $5y^2-20y=0$, so $y=0$ or $y=4$
3. Find the matching $x$ values and give coordinates: $(-5,0)$ and $(3,4)$

True or false? Correct values found by trial and improvement, with no algebra shown, earn the marks in a 'form and solve' question.

False. An answer from trial and improvement with no working scores no marks; the equation must be formed and solved.