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Chapter 10: Problem 74

When you multiply a binomial containing a square root by its conjugate, what happens to the radical?

Short Answer

Expert verified
When you multiply a binomial containing a square root by its conjugate, the terms with the square root cancel each other out, leaving a real number result without any square root term. In general, \((a + b\sqrt{x})(a - b\sqrt{x}) = a^2 - b^2x\), where the radical has been eliminated.

Step by step solution

01

Write down the given binomial and its conjugate

Consider the binomial \(a + b\sqrt{x}\), where 'a', 'b', and 'x' are real numbers. The conjugate of this binomial is \(a - b\sqrt{x}\).
02

Multiply the binomial and its conjugate

To multiply the binomial and its conjugate, we can use the FOIL (First, Outer, Inner, Last) method. This simply means we multiply the first terms, then the outer terms, then the inner terms, and finally the last terms, and then add the results together. So, we have \((a + b\sqrt{x})(a - b\sqrt{x})\).
03

Apply the FOIL method

Using the FOIL method, we get: - First terms: \(a \cdot a = a^2\) - Outer terms: \(a \cdot (-b\sqrt{x}) = -ab\sqrt{x}\) - Inner terms: \(b\sqrt{x} \cdot a = ab\sqrt{x}\) - Last terms: \(b\sqrt{x} \cdot (-b\sqrt{x}) = -b^2x\) Now, sum these results: \(a^2 - ab\sqrt{x} + ab\sqrt{x} - b^2x\).
04

Observe what happens with the radical

In the sum calculated in the previous step, we can see that the terms containing the square root of 'x' cancel each other out: \(- ab\sqrt{x} + ab\sqrt{x} = 0\). This leaves us with the result: \(a^2 - b^2x\). Thus, when multiplying a binomial containing a square root by its conjugate, the radical disappears and we are left with a real number result, without any square root term.

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