Square Roots and the Equations They Solve

The symbol in $\sqrt{49}$ is called the radical sign. You read $\sqrt{49}$ as the square root of 49.

Here is what it means. For any number $n \ge 0$, $\sqrt{n}$ is the nonnegative number whose square is $n$. Nonnegative means zero or positive.

So, for $n \ge 0$, a number $r$ equals $\sqrt{n}$ exactly when it passes two checks: $r \ge 0$ and $r^2 = n$.

• $\sqrt{49} = 7$, because $7 \ge 0$ and $7^2 = 49$.
• $\sqrt{0} = 0$, because $0 \ge 0$ and $0^2 = 0$.

When $n$ is a perfect square, $\sqrt{n}$ is a whole number. Every number under a radical sign in this lesson is a perfect square, so each root you find is an exact value, not an estimate.

The table shows the perfect squares from $1$ to $144$. Each row gives a whole number, its square, and the square root fact that goes with it.

Why does the meaning of $\sqrt{n}$ insist on a nonnegative number? Because $7$ is not the only number whose square is $49$. A negative number times a negative number is positive, so $(-7)^2 = (-7)(-7) = 49$.

So $7$ and $-7$ are both square roots of $49$. They are opposites: two numbers the same distance from zero, on opposite sides of it.

Three symbols keep these roots apart:

• $\sqrt{49} = 7$ is the principal square root, the nonnegative one. For $n \ge 0$, the value of $\sqrt{n}$ is never negative.
• $-\sqrt{49} = -7$ is the negative square root. The minus sign sits outside the radical sign, so the number under it is still $49$.
• $\pm\sqrt{49}$ is read plus or minus the square root of 49. It names both roots at once: $7$ and $-7$.

Does every number have two square roots? No. It depends on whether the number is positive, zero or negative. Here, number means a real number: any number on the number line.

• A positive number has exactly two real square roots, and they are opposites. Among nonnegative numbers, a larger number has a larger square: $6^2 = 36$, $7^2 = 49$, $8^2 = 64$. So only one nonnegative number, $7$, has the square $49$. A number and its opposite have equal squares, so on the negative side only $-7$ works.
• Zero has exactly one square root, which is $0$. You know that $0^2 = 0$, and every other real number has a positive square.
• A negative number has no real square root. A positive number times itself is positive, a negative number times itself is positive, and $0 \cdot 0 = 0$. So the square of a real number is never negative. That is why the meaning of $\sqrt{n}$ requires $n \ge 0$.

Now you can use square roots to solve equations. An equation says that two expressions are equal. A solution of an equation is a value of the variable that makes it true.

Take $x^2 = 49$, where the variable $x$ stands for a real number. The equation asks: which real numbers have the square 49? That is the same question as what are the square roots of 49? You already know them: $7$ and $-7$. So the equation has two solutions, $x = 7$ and $x = -7$. You can write them together as $x = \pm 7$.

The same reasoning works for any equation of the form $x^2 = n$: its real solutions are exactly the real square roots of $n$. So before you solve, compare $n$ with zero to see how many solutions to expect.

Takeaway. For $n \ge 0$, the symbol $\sqrt{n}$ names one number: the nonnegative principal square root, so $\sqrt{49} = 7$. The equation $x^2 = n$ asks for every real number whose square is $n$. It has two solutions, $x = \pm\sqrt{n}$, when $n > 0$; one solution, $x = 0$, when $n = 0$; and no real solution when $n < 0$. When a situation limits the variable, as a length does, keep only the solutions that fit.

Next, you will work backward from cubes such as $2^3 = 8$: you will find cube roots and use them to solve equations.