3.4 Radical equations & functions

Radical Equations and Functions

A radical uses the root symbol √, and the expression under it is the radicand. This section connects radicals to rational exponents, shows how to solve radical equations safely, and describes the graphs and domains of square-root functions.

Radicals and Rational Exponents

A fractional exponent is another way to write a root. In general, x^(1/n) is the nth root of x, and x^(m/n) equals the nth root of x raised to the m power.

Reading the fraction is the whole skill: the bottom number of the exponent is the root, and the top number is the power. Order does not matter, so you may take the root first and then the power, whichever keeps the numbers small. For instance, 32^(2/5) equals the fifth root of 32 (which is 2) raised to the second power, so 2² = 4. Taking the root first avoids ever computing the large number 32² = 1024.

Example 1 (simplify a radical). √50 = √(25 · 2) = √25 · √2 = 5√2. Similarly, √72 = √(36 · 2) = 6√2. The trick is to pull out the largest perfect-square factor so what remains under the radical has no perfect-square factors left.

Solving Radical Equations

To solve an equation with a square root, follow three steps: isolate the radical, square both sides, and check for extraneous roots. Squaring can create false solutions, so the check is required, not optional.

Example 2. Solve √(2x + 1) = 3. The radical is already isolated. Square both sides: 2x + 1 = 9. Then 2x = 8, so x = 4. Check: √(2·4 + 1) = √9 = 3. Correct.

Example 3 (isolate first). Solve √(x − 1) + 2 = 5. Subtract 2 to isolate the radical: √(x − 1) = 3. Square both sides: x − 1 = 9, so x = 10. Check: √(10 − 1) + 2 = √9 + 2 = 3 + 2 = 5. Correct.

Example 4 (extraneous root). Solve √x = x − 6. Square both sides: x = (x − 6)² = x² − 12x + 36. Move everything to one side: 0 = x² − 13x + 36. Factor: (x − 4)(x − 9) = 0, so x = 4 or x = 9. Now check each in the original equation. For x = 9: √9 = 3 and 9 − 6 = 3, so both sides equal 3. Valid. For x = 4: √4 = 2 but 4 − 6 = −2, and 2 ≠ −2. This is extraneous. The only solution is x = 9.

Example 4 shows exactly why you must check: the algebra offered two candidates, but only one actually satisfies the original equation. A common mistake in that step is squaring the right side incorrectly. Remember that (x − 6)² means (x − 6)(x − 6) = x² − 12x + 36, not x² + 36 — you must include the middle term. Whenever you square a two-term expression, expand it fully before solving.

Radical (Square-Root) Functions

A square-root function such as f(x) = √x has a graph that starts at a point and curves gently upward to the right, rising more and more slowly. Because you cannot take the square root of a negative number in the real number system, the domain is restricted: the radicand must be greater than or equal to zero.

Example 5 (domain from the radicand). For f(x) = √(x − 4), set the radicand at or above zero: x − 4 ≥ 0, so x ≥ 4. The domain is all real numbers x ≥ 4, and the range is y ≥ 0.

Example 6 (another domain). For f(x) = √(2x − 6), require 2x − 6 ≥ 0, so 2x ≥ 6 and x ≥ 3. The domain is x ≥ 3.

Example 7 (a decreasing radicand). For f(x) = √(10 − 2x), require 10 − 2x ≥ 0, so 10 ≥ 2x and x ≤ 5. Here the inequality flips the usual direction because the variable is subtracted, giving the domain x ≤ 5.

Example 8 (evaluate). If f(x) = √(x + 5), then f(4) = √(4 + 5) = √9 = 3.

The range of a basic square-root function is y ≥ 0, because the principal square root never returns a negative number. On a graph, the curve begins at the point where the radicand equals zero and then rises to the right, never dropping below the x-axis. Knowing that the output cannot be negative also helps you spot extraneous roots quickly: if solving forces the square root to equal a negative number, there is no real solution.

Here is a quick reference of perfect powers that speeds up radical work:

Memorizing these squares and cubes lets you recognize radicals and rational exponents on sight. Combine that recall with the isolate-square-check routine, and remember to test the domain whenever a variable sits under a root — those habits turn most TSIA2 radical questions into quick, reliable points.