Conjugates

Conjugates are binomial pairs that have the same numbers/variables within them, but opposite signs. So for example (x-1) and (x+1) are conjugates. Conjugates have special properties that can be useful to know when they appear in questions or answers on the ACT.

Simplifying and Rationalizing Denominators

This particularly true when the original denominator contains either a radical (square root) or imaginary numbers (i = sqrt(-1)).

For instance, if the fraction 

510+3iis given, then this can be multiplied by 

10–3i10–3i(standard algebra rule: multiplying it by one does not change the value of the fraction).

This leads to the fraction: 

50–15i100–3i2. i² =  − 1, so  − 3i²= + 3. So the denominator of the fraction can be simplified to 103, giving the final fraction 

50–15i103.

Imaginary roots of polynomials always come in conjugate pairs. So whenever you see a binomial that includes i, think conjugate.

Conjugates and Quadratics

When conjugates are factors of quadratic equations they cancel out the middle terms of the equation and leave only a² + c. Whenever this equation appears in a problem, it means that the equation can be factored into a conjugate pair.