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Lesson · Grade 11 · Pre Calculus

Solving a Radical Equation and Checking for Extraneous Roots

  • Grade 11
  • Pre Calculus
  • 2 min 7 s
  • British Columbia
  • BC11-MA-67

Students solve √(x + 3) = x − 3 by finding the domain, squaring both sides and factoring x² − 7x + 6 = 0, which gives x = 1 or x = 6. Checking both in the original equation, and on a graph of both sides, shows that only x = 6 works, because squaring added the extraneous root x = 1.

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In one sentence

The question this lesson answers

Let us solve: the square root of x plus 3 equals x minus 3.

With x under a square root sign, this is called a radical equation.

Transcript

What the lesson says

Let us solve: the square root of x plus 3 equals x minus 3.

With x under a square root sign, this is called a radical equation.

The domain is the values x may take. Under the root, x plus 3 cannot be negative, so x is at least negative 3.

A square root is never negative, so x minus 3 cannot be either: x is at least 3.

Now square both sides to remove the root.

x minus 3, all squared, is x squared minus 6 x plus 9.

Take x and 3 from both sides: x squared minus 7 x plus 6 equals 0.

Factor it. Negative 1 times negative 6 is 6, and negative 1 plus negative 6 is negative 7.

So x is 1 or 6, but these are only possible answers until we check them.

Squaring can add answers that do not fit the original. They are called extraneous roots.

Check 6 in the original. The square root of 9 is 3, and 6 minus 3 is 3. They match.

Now check 1. The square root of 4 is 2, but 1 minus 3 is negative 2, so 1 is an extraneous root.

Squaring hides the sign: 2 squared and negative 2 squared are both 4.

Graph y equals the square root of x plus 3, and y equals x minus 3.

They meet only once, at the point 6, 3, so x equals 6 is the only solution.

At x equals 1, the curve is at 2, but the line is at negative 2.

A common mistake is skipping the check. Squaring can add answers.

Recap: square both sides, solve, then check every answer in the original.

Curriculum

Where this fits

British Columbia, Grade 11

Pre Calculus 11, Real numbers and powers: radical operations and equationsBC11-MA-67

Previous: Simplifying, Adding and Multiplying Radicals

The outcomes and strands follow the British Columbia curriculum.

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