Before this page: be able to square a whole number using index notation, and know the square root as the reverse of squaring from square roots.
Two numbers, the same square
Suppose you square a number and get 49. Can you work backwards to find the number you started with? 7 works, but it isn't the only possibility.
7 × 7 = 49Two positive factors
(−7) × (−7) = 49Two negative factors
Does this work for other numbers too? Choose a number on the line, then change its sign.
Squaring gives the same result for a number and its negative, so knowing the square doesn't tell us which sign we started with.
What the square-root symbol means
7 and −7 are both square roots of 49. But the square-root symbol has a more specific meaning: it asks for the non-negative root. This is called the principal square root: the positive root, or zero when the number is zero. The symbol therefore has one definite value.
49 = 7
We write the negative root as −49.
The minus sign changes the sign of the answer, not the meaning of the root symbol. In these two examples, we first find the principal root of 36; only then do we apply the minus sign, if there is one.
Finding both solutions
What if we ask which numbers square to 36, instead of asking for its principal root? The equation x2 = 36 has both 6 and −6 as its solutions.
x = ±6 is a shorter way to write “x = 6 or x = −6”. It gives both values of the unknown. We read ±6 as “plus or minus 6”.
Example: solve x2 = 81
Find both numbers that square to 81, then record them on one line.
- x2 = 81 81 is positive, so two numbers square to it.
- 9 × 9 = 81 9 works. So does −9, because two negative factors have a positive product.
- x = ±9 Both are solutions, and ±9 records them on one line.
Can a square be zero or negative?
x2 = 0 has one solution, 0, because 0 and −0 are the same number.
Could a negative number work for a negative result? No: multiplying two negatives gives a positive. No number on the number line squares to −9. A square is a number multiplied by itself, so the two numbers being multiplied are identical. Their product is positive or zero.
- x2 = 49 Two solutions, x = ±7
- x2 = 0 One solution, x = 0
- x2 = −9 No number on the line
Square roots and solutions
Key points
- The square-root symbol gives the non-negative root. A minus sign outside it takes the negative of that value.
- When solving an equation, include every value that works. The ± sign is shorthand for a positive value and its negative.
- A positive number has two square roots; zero has one. No number on the number line has a negative square.