Mathematics
What Is a Square Root?
√25 is 5, but solving x² = 25 gives both 5 and −5. That distinction costs marks every year, and it takes about a paragraph to understand properly.
Square root
A square root of a number is a value that gives that number when multiplied by itself; the sign √ denotes the positive one, so √25 = 5.
- Where students meet it
- Grade 6 to 8; in England, using integer powers and their associated real roots is a key stage 3 objective, after which roots appear in Pythagoras, quadratics and trigonometry.
The short answer
A square root of a number is a value that multiplies by itself to give that number, so √25 is 5 because 5 × 5 = 25. The √ sign means the positive root only. The equation x² = 25 is a different question and has two answers, 5 and −5.
An example
√49 = 7, but x² = 49 gives x = 7 or x = −7
Both lines are correct and they are not in conflict. The first evaluates a symbol that has been defined to return the positive root. The second solves an equation, which means finding every value that satisfies it, and (−7) × (−7) is also 49. Losing the negative solution is one of the most reliable sources of dropped marks in quadratics.
Two numbers square to 25; the symbol picks one
Both 5 and −5 give 25 when squared, so 25 has two square roots. The symbol √ was defined to return the non-negative one, because a symbol that produced two answers at once would be unusable in the middle of a longer expression.
So √25 is 5, full stop, and if the negative root is wanted it is written −√25. When an equation such as x² = 25 is being solved, no symbol is doing the choosing and both values qualify, which is what the ± in x = ±5 records. Reading the two situations as one is what produces half-marks year after year.
The roots worth recognising instantly
Knowing the squares forwards is Grade 5 work. Knowing them backwards is what makes simplifying surds, factorising quadratics and checking a Pythagoras answer quick rather than laborious, and the useful range stops at about 15.
It is worth rehearsing in both directions, so that a student shown 169 answers 13 rather than reaching for a calculator.
- √1 = 1, √4 = 2, √9 = 3, √16 = 4, √25 = 5
- √36 = 6, √49 = 7, √64 = 8, √81 = 9, √100 = 10
- √121 = 11, √144 = 12, √169 = 13, √196 = 14, √225 = 15
Estimating √50 without a calculator
Find the perfect squares on either side. 49 and 64 bracket 50, so √50 sits between 7 and 8. Since 50 is barely past 49, the answer is only just past 7 — about 7.07, though the estimate a non-calculator paper wants is usually just over 7. Checking is quick: 7 squared is 49 and 7.1 squared is 50.41, so the answer sits between them.
The same bracketing handles anything. √30 lies between √25 and √36, so between 5 and 6, and nearer 5.5 than either end. A question asking you to estimate is testing exactly this, and the two squares you name in your working are what earns the method mark.
Common questions
Is √25 equal to −5?
No. √25 is 5. Both 5 and −5 square to 25, so 25 has two square roots, but the √ symbol is defined to give the positive one. If the negative is wanted it must be written with a minus sign in front: −√25 = −5.
What is the square root of a negative number?
There is no real answer. A positive squared is positive and a negative squared is also positive, so nothing real squares to −16. Beyond school level a new kind of number is introduced to handle this, but at GCSE and IGCSE the correct response is that the expression has no real value.
What is a surd?
A root that cannot be written exactly as a fraction or a terminating decimal, so it is left in root form. √2 and √3 are surds; √9 is not, since it is 3. Leaving an answer as 3√2 rather than 4.24 keeps it exact and stops rounding errors entering later working.
How do I estimate a square root without a calculator?
Name the perfect squares immediately below and above. For √85, that is 81 and 100, so the answer lies between 9 and 10 and, since 85 is much closer to 81, near 9.2. Writing down both bracketing squares is what shows the method to an examiner.
Sources
- National curriculum in England: mathematics programmes of study — Department for Education
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