What is the difference between a square root and a solution to x squared equals a number?
Quick answer
The symbol √ means the principal, nonnegative square root: √25 = 5. But the equation x² = 25 asks which values of x square to 25. Both 5 and −5 do, so the equation has two solutions. For a positive number k, solving x² = k gives x = ±√k; the ± belongs to the equation-solving step, not to the value of √k.
The question, the details, the next step.
I keep seeing √25 = 5 and x² = 25 gives x = ±5. Aren’t those statements contradicting each other?
They answer different questions. The expression √25 asks for the principal square root, which is the nonnegative value whose square is 25. So √25 = 5, not ±5. The equation x² = 25 asks for every value that makes the equation true; both 5 and −5 work. [S1]
A useful comparison is: evaluate the radical → one nonnegative value; solve the equation → list all values that satisfy it. The plus-or-minus appears in the second task because there are two solutions for this positive square number.
Can you show the two steps side by side so I can see where the ± comes from?
Sure. Use the same number, 25:
- Evaluate the radical: √25 = 5. The radical names the principal, nonnegative root. [S1]
- Solve the equation: x² = 25, so x = ±√25 = ±5. Therefore x = 5 or x = −5. For x² = k with k ≥ 0, the square-root property gives x = ±√k. [S2]
Check each candidate in the original equation: 5² = 25, and (−5)² = 25. Both satisfy it. The minus sign does not make √25 negative; it gives the second value whose square is 25.
When I’m doing homework, how can I tell which form the question wants?
Look at what is being asked. If the task is to simplify or evaluate a radical such as √25, give the principal value, 5. If it asks you to solve x² = 25, find all values of x that make the equation true, so give x = 5 and x = −5. A compact way to remember the distinction is: √25 is a value; x² = 25 is a condition to solve.
For an equation x² = k with k ≥ 0, the square-root property is written x = ±√k. [S2] The ± is part of the solving rule. It does not change the definition of the radical symbol.
What if I’m unsure whether I’ve included too many answers?
Substitute each proposed value into the original equation, rather than judging by the radical notation alone. For x² = 25, test x = 5 and x = −5: each square equals 25, so both belong in the solution set. For the separate expression √25, there is no equation asking you to collect candidates; its value is just 5. [S1]
A quick written solution can make the distinction visible:
Radical: √25 = 5.
Equation: x² = 25 → x = ±√25 → x = 5 or x = −5.
The examples establish this distinction and the square-root property for nonnegative k; they do not address every possible equation or every context in which a radical appears.
A practical checklist
Decide whether the prompt asks you to evaluate a radical or solve an equation.
For √25, write the principal, nonnegative value: 5.
For x² = 25, use x = ±√25 and list both candidates.
Substitute each candidate into the original equation to check it.
Keep the ± attached to solving the equation, not to the value of √25.
A few more useful answers
Should I write √25 = ±5?
Why does −5 work in x² = 25?
Squaring −5 gives 25, so it satisfies the equation just as 5 does. Check proposed values in the original equation.
Where does the plus-or-minus sign go when solving x² = 25?
Use x = ±√25, then write the two solutions, x = 5 or x = −5. The ± represents those two equation solutions. [S2]
Sources
References used for this guide.
- [S1] OpenStax College Algebra 2e: Principal Square Roots openstax.org · Checked
- [S2] OpenStax College Algebra 2e: Square Root Property openstax.org · Checked