The Principal Variable

In one sentence

in x : cubic x^{3} - a x^{2} + 2 x - 2 a = in a : linear - (x^{2} + 2) a + (x^{3} + 2 x)

When an expression has several letters, problems usually call $x$ “the unknown” and the others “parameters”. But that is only a name: the expression itself does not know which letter is the unknown. Pick the letter with the lowest degree as the main variable, treat every other letter as part of the coefficients, and the problem often becomes simple at once.

Example: factor x³ − ax² + 2x − 2a

Step 1 / 4

The usual view
Ordered by powers of $x$ : a cubic, with $a$ mixed into the coefficients. No ready-made formula applies.

$x^{3} - a x^{2} + 2 x - 2 a$
Change the main variable
Look at the two coefficients
Take it out

Predict first

$x^{4} + x^{2} + 2 a x + 1 - a^{2}$ has degree four in $x$ . What is its degree in $a$ ?

OneTwoFour

Choose an answer first; then you can see the result.

Stop 1 · Factoring quadratics in two variables (Vol. 1)

When both letters appear squared, pick either one as the main variable and the other goes into the coefficients: it is the cross method, with expressions as coefficients.

Example: factor x² + 3xy + 2y² + 4x + 5y + 3

Step 1 / 4

Make x the main variable
Group the terms into three piles: those with $x^{2}$ , those with $x$ , and those with no $x$ .

$x^{2} + (3 y + 4) x + (2 y^{2} + 5 y + 3)$
Factor the “constant term” first
Match the “coefficient of x”
Write the result

Stop 2 · Equations with letter coefficients (Vol. 2)

“Holds for every value of $m$ ” really says: with $m$ as the main variable, you have an identity in $m$ . A linear expression $A m + B$ equals $0$ for every $m$ only if $A = 0$ and $B = 0$ .

Example: for every value of m, the line (m+1)x + (m−2)y − 3m = 0 passes through the same point. Find it

Step 1 / 5

What is changing?
$m$ can take any value, while the point $(x, y)$ we want is fixed. So the real “variable” here is $m$ .

$(m + 1) x + (m - 2) y - 3 m = 0$
Make m the main variable
True for every m
Solve the system
Check

Stop 3 · Integer roots of quadratic equations (Vol. 2)

The equation is quadratic in $x$ , but linear in the parameter $a$ . So solve for $a$ , and let divisibility pick out $x$ .

Example: find all integers a for which x² − ax + a + 3 = 0 has an integer root

Step 1 / 5

The usual view
With $x$ as the main variable, the discriminant $a^{2} - 4 a - 12$ has to be a perfect square. That works, but it is the long way round.

$Δ = a^{2} - 4 a - 12$
Change the main variable
Do one division
Let divisibility pick x
Work out a

Stop 4 · Finding a range (Vol. 2 · the discriminant)

To find how large $x$ can be, you actually treat $x$ as a known coefficient and make the other letter the main variable: whether $x$ can take a certain value depends on whether the equation in $y$ has a real root.

Example: real numbers x and y satisfy x² + xy + y² = 3. Find the maximum value of x

Step 1 / 4

Make y the main variable
Treat $x$ as a coefficient. This is a quadratic equation in $y$ .

$y^{2} + x y + (x^{2} - 3) = 0$
What “x can take this value” means
Work out the discriminant
Read off the answer

Practice

Exercise 1Basic

For every value of $k$ , the line $y = k x - 2 k + 3$ passes through the same point. What is the sum of that point's $x$ - and $y$ -coordinates?

Exercise 2Advanced

Find all integers $a$ for which the equation $x^{2} - a x + a + 1 = 0$ (in $x$ ) has an integer root. (Separate them with a comma.)

Exercise 3Challenge

Real numbers $x$ and $y$ satisfy $x^{2} - x y + y^{2} = 12$ . Find the maximum value of $x$ .

In one sentence

Example: factor x³ − ax² + 2x − 2a

Stop 1 · Factoring quadratics in two variables (Vol. 1)

Example: factor x² + 3xy + 2y² + 4x + 5y + 3

Stop 2 · Equations with letter coefficients (Vol. 2)

Example: for every value of m, the line (m+1)x + (m−2)y − 3m = 0 passes through the same point. Find it

Stop 3 · Integer roots of quadratic equations (Vol. 2)

Example: find all integers a for which x² − ax + a + 3 = 0 has an integer root

Stop 4 · Finding a range (Vol. 2 · the discriminant)

Example: real numbers x and y satisfy x² + xy + y² = 3. Find the maximum value of x

Practice

Mistake Museum · Exhibits from this stop

Dividing by a coefficient that contains a letter

Stops on this line