Square Root Formula
The square root formula is the definition itself: the square root of a number x is the value y such that y² = x, written y = √x. The formula earns its keep in equations, where taking the square root of both sides undoes a square: if x² = 49, then x = ±7, because both 7 and −7 square to 49. The square root calculator above evaluates any root instantly, and this page shows how the formula works inside equations, geometry, and the quadratic formula.
The most important use of the square root formula is the quadratic formula, which solves any equation of the form ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) / 2a. The square root in the middle decides how many solutions there are: a positive value under the root gives two solutions, zero gives one, and a negative value gives none in real numbers. This page works through that formula and the simpler square root equations that lead up to it.
The Square Root: What It Is and How to Find It
The square root of a number is the value that, multiplied by itself, gives the original number. The square root of 144 is 12, because 12 × 12 = 144, and the square root of 2 is about 1.41421, because 1.41421 × 1.41421 is very close to 2. The symbol is √, so √144 = 12 and √2 ≈ 1.41421. Every positive number has two square roots, one positive and one negative, but the symbol √ on its own always means the positive one, called the principal square root. The square root of a negative number is not a real number at all, which is why the calculator reports an error for negative inputs.
Most square roots are irrational numbers: their decimal expansion never ends and never repeats. The square root of 2 is the famous example — its digits run 1.41421356 and continue forever without a pattern, a fact proved by the ancient Greeks. That is why we round: √50 ≈ 7.07 is accurate to two decimal places, and √50 ≈ 7.071 is accurate to three. The calculator above also handles cube roots and other roots: the cube root of 27 is 3, because 3 × 3 × 3 = 27, and the fourth root of 81 is 3, because 3⁴ = 81.
Square roots appear wherever a square is involved: the side of a square from its area, the diagonal of a rectangle, the standard deviation in statistics, and the quadratic formula in algebra. The National Institute of Standards and Technology lists the square roots of small integers among its standard mathematical constants, and the same values turn up in geometry, physics, and engineering problems of every kind.
Step 1 — Check Whether the Number Is a Perfect Square
Try the squares you know: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. If your number is on that list, its square root is exact: √144 = 12, √225 = 15. If it is not, the root is irrational and you will estimate or use the calculator.
Step 2 — Estimate Between Two Whole Numbers
For √50, note that 49 is 7² and 64 is 8², so √50 sits just above 7. For √125, note that 121 is 11² and 144 is 12², so the answer sits just above 11, and indeed √125 ≈ 11.18. A good estimate tells you at a glance whether a calculator answer is plausible.
Step 3 — Refine and Check by Squaring
Refine the estimate by squaring your guess: 7.07² = 49.9849, just under 50, so √50 ≈ 7.07 is right to two decimal places. Always check by squaring the answer you report; if the square does not return your original number, something went wrong.
Worked Examples: The Square Root Formula in Equations
Each row solves an equation by taking the square root of both sides, with a check by substitution.
| Equation | Square root step | Solution |
|---|---|---|
| x² = 49 | x = ±√49 | x = 7 or x = −7 |
| x² = 144 | x = ±√144 | x = 12 or x = −12 |
| x² = 2 | x = ±√2 | x = 1.41421 or x = −1.41421 |
| x² − 5x + 6 = 0 | x = (5 ± √(25 − 24)) / 2 | x = 3 or x = 2 |
| x² = 50 | x = ±√50 | x = 7.07 or x = −7.07 |
The quadratic example shows the formula in action: for x² − 5x + 6 = 0, a = 1, b = −5, and c = 6, so the discriminant is b² − 4ac = 25 − 24 = 1, and √1 = 1. The formula gives x = (5 + 1) / 2 = 3 and x = (5 − 1) / 2 = 2, and both check out: 3² − 15 + 6 = 0, and 2² − 10 + 6 = 0. The discriminant is the number under the root, and its sign tells you how many real solutions exist before you finish the calculation.
How to Use the Square Root Calculator
Enter any number, choose the root, and the result appears instantly, with the exact simplified form where one exists. Use it for homework checks, geometry, and any calculation where a root is one step of a longer problem.
- Enter the number. Type the value whose root you want, such as 72. The default of 72 shows √72 ≈ 8.49, or 6√2 in exact form.
- Choose the root. Square root is the default; switch to cube root, fourth root, or any other root for problems such as ∛27 = 3.
- Read the decimal result. The calculator gives the root to several decimal places, which is the form you need for measurement and engineering work.
- Read the simplified form. Where the number has a square factor, the calculator shows the exact form: √72 = 6√2, √50 = 5√2, √125 = 5√5.
- Square the answer to check. Multiply the result by itself and confirm you get back to the number you entered; that one check catches almost every input error.
The Quadratic Formula, Step by Step
The quadratic formula is the square root formula applied to a general quadratic equation, and its parts each have a meaning. This table breaks it down.
| Part | Formula piece | What it means |
|---|---|---|
| a, b, c | Coefficients of ax² + bx + c = 0 | The numbers that define the equation |
| Discriminant | b² − 4ac | Decides how many solutions exist |
| Square root | √(b² − 4ac) | The spread between the two solutions |
| ± | Plus or minus | Two solutions from one formula |
| Denominator | 2a | Scales the result back to x |
Common Mistakes With Square Roots
Square roots are a classic source of small, stubborn errors. These are the ones to watch for.
- Adding roots as if they were numbers. √2 + √3 is not √5. Roots add only when the radicand matches: √2 + 3√2 = 4√2.
- Forgetting the negative root. The equation x² = 49 has two solutions, x = 7 and x = −7, even though √49 on its own means 7.
- Taking the root of a negative number. There is no real number whose square is negative, so √(−4) has no real answer and the calculator correctly refuses it.
- Rounding too early. Rounding √2 to 1.41 and then squaring gives 1.9881, not 2. Keep extra digits until the last step of the problem.
- Confusing square and square root. Squaring makes numbers bigger than 1 larger; the square root makes them smaller. √144 = 12, while 144² = 20,736.
Where You Will Use Square Roots
Square roots are not just a school topic. They appear in practical work wherever area, distance, or variation is measured.
- Geometry. The side of a square garden with an area of 144 square metres is √144 = 12 metres, and the diagonal of a 3 by 4 rectangle is √(9 + 16) = 5.
- Statistics. The standard deviation is the square root of the variance, so every report of spread in test scores or measurements runs through a square root.
- Physics. The speed of a falling object, the period of a pendulum, and the energy in a spring all involve square roots of the quantities you measure.
- Construction. The 3-4-5 triangle, checked with √(3² + 4²) = 5, is how builders square up corners on site.
- Finance. The square root of time rule scales risk and volatility: over four times the period, typical movement roughly doubles, because √4 = 2.
The Discriminant: How Many Solutions?
The discriminant, b² − 4ac, is the number under the square root in the quadratic formula, and its sign is the whole story. If it is positive, the square root is real and there are two different solutions, as in x² − 5x + 6 = 0, where the discriminant is 1. If it is zero, the square root is zero and there is exactly one solution, a repeated root. If it is negative, the square root has no real value and the equation has no real solutions, though it has two complex ones. Checking the discriminant first saves the work of finishing a calculation that has no real answer.
Tips for Accurate Results
A little care with roots pays off, because a root error hides inside every later step of a problem.
- Memorise the squares up to 15². Knowing 12² = 144 and 15² = 225 lets you spot perfect squares instantly and estimate everything else.
- Simplify before approximating. Reducing √72 to 6√2 keeps the answer exact, and the decimal 8.49 follows from it without extra rounding error.
- Keep two extra digits. Carry two more decimal places than your final answer needs, and round only at the end.
- Check with a square. Squaring your answer is the fastest possible check, and it takes two seconds.
- Watch the domain. The input must be zero or positive for a real square root; a negative input means the problem needs a different approach.
Square Root Quick Answers
These are the root values people look up most often, each one verified against the calculator above.
| Question | Answer |
|---|---|
| Square root of 144 | 12 |
| Square root of 50 | 7.07 |
| Square root of 2 | 1.41421 |
| Cube root of 27 | 3 |
| Cube root of 64 | 4 |
Related tools on this site: the main square root calculator, scientific calculator, basic calculator, average calculator, and the square root calculator again whenever you need a fresh check.
Frequently Asked Questions
What is the square root formula?
The square root of x is the value y such that y² = x, written y = √x. In equations, taking the square root of both sides undoes a square: if x² = 49, then x = ±√49, so x = 7 or x = −7.
How do I solve x² = 49?
Take the square root of both sides: x = ±√49, so x = 7 or x = −7. Both values check out, because 7² = 49 and (−7)² = 49. Forgetting the negative solution is the most common error with this equation.
What is the quadratic formula?
For ax² + bx + c = 0, the solutions are x = (−b ± √(b² − 4ac)) / 2a. The square root in the middle is the discriminant, and its sign tells you whether there are two, one, or no real solutions.
What does the discriminant tell me?
The discriminant is b² − 4ac. If it is positive, there are two real solutions; if it is zero, there is one repeated solution; if it is negative, there are no real solutions. For x² − 5x + 6 = 0, it is 1, so there are two solutions, 2 and 3.
How do I solve x² = 2?
Take the square root of both sides: x = ±√2, so x ≈ 1.41421 or x ≈ −1.41421. Both values square to 2, and the answers are irrational, so they are rounded to the digits you need.
Disclaimer: This calculator is for general information only. Results are rounded for display; for engineering, construction, or safety-critical work, confirm measurements with the applicable standards and a qualified professional.