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Conjugates

Conjugates

The result of writing sum of two terms as a difference or vice-versa. Note: Conjugates are similar to, but not the same as, complex conjugates.

 

Expression Conjugate
x + 2
6 minus square root of 7
-4 + 3√5
The expression: 1 minus cosine theta
x – 2
6 plus square root of 7
The expression negative 4 minus 3 times the square root of 5
The expression 1 + cos θ

Key Formula

(a+b)(a−b)=a2−b2\left(a + b\right)\left(a - b\right) = a^2 - b^2
Where:
  • aa = The first term in each binomial
  • bb = The second term whose sign changes between the two conjugates

Worked Example

Problem: Rationalize the denominator of the fraction 53+2\dfrac{5}{3 + \sqrt{2}}.
Step 1: Identify the conjugate of the denominator. The denominator is 3+23 + \sqrt{2}, so its conjugate is 3−23 - \sqrt{2}.
3+2⟶3−23 + \sqrt{2} \longrightarrow 3 - \sqrt{2}
Step 2: Multiply both numerator and denominator by the conjugate. This does not change the value of the fraction because you are multiplying by 1.
53+2⋅3−23−2\frac{5}{3 + \sqrt{2}} \cdot \frac{3 - \sqrt{2}}{3 - \sqrt{2}}
Step 3: Multiply out the denominator using the difference of squares pattern: (a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 - b^2.
(3)2−(2)2=9−2=7(3)^2 - (\sqrt{2})^2 = 9 - 2 = 7
Step 4: Multiply out the numerator by distributing.
5(3−2)=15−525(3 - \sqrt{2}) = 15 - 5\sqrt{2}
Step 5: Write the final simplified fraction.
15−527\frac{15 - 5\sqrt{2}}{7}
Answer: 15−527\dfrac{15 - 5\sqrt{2}}{7}

Another Example

Problem: Multiply the conjugate pair (x+7)(x−7)(x + 7)(x - 7).
Step 1: Recognize that these are conjugates: same terms, opposite signs. Apply the difference of squares formula with a=xa = x and b=7b = 7.
(x+7)(x−7)=x2−72(x + 7)(x - 7) = x^2 - 7^2
Step 2: Simplify.
x2−49x^2 - 49
Answer: x2−49x^2 - 49

Frequently Asked Questions

Why does multiplying conjugates eliminate the middle term?
When you expand (a+b)(a−b)(a+b)(a-b) using FOIL, the two middle terms are −ab-ab and +ab+ab, which cancel each other out. You are left with only a2−b2a^2 - b^2, the difference of squares. This cancellation is exactly what makes conjugates useful.
What is the difference between a conjugate and a complex conjugate?
A conjugate (sometimes called a radical conjugate or binomial conjugate) swaps the sign between any two terms, such as 5+35 + \sqrt{3} becoming 5−35 - \sqrt{3}. A complex conjugate specifically changes the sign of the imaginary part: the complex conjugate of a+bia + bi is a−bia - bi. The idea is the same—flip one sign—but complex conjugates apply only to complex numbers.

Conjugate (binomial) vs. Complex conjugate

Why It Matters

Conjugates are the key tool for rationalizing denominators—removing radicals from the bottom of a fraction—which is a standard requirement in algebra. They also appear in factoring when you recognize a difference of squares, letting you factor expressions like x2−25x^2 - 25 into (x+5)(x−5)(x+5)(x-5). In more advanced courses, the same sign-flipping idea extends to complex conjugates used in solving polynomial equations and simplifying complex expressions.

Common Mistakes

Mistake: Changing both signs instead of only the sign between the two terms.
Correction: Only the sign connecting the two terms flips. The conjugate of −x+3-x + 3 is −x−3-x - 3, not x−3x - 3. You change the ++ between the terms to a −-, leaving the signs of the individual terms alone.
Mistake: Trying to find a conjugate of a trinomial (three-term expression).
Correction: Conjugates apply to binomials (two-term expressions). If you have three or more terms, you cannot simply swap one sign and expect the middle terms to cancel. You may need to group terms first, for example treating (a+b)+c(a + b) + c as a binomial with grouped first term.

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