Square Roots

Definition

A square root is a mathematical operation that finds the value that, when multiplied by itself, equals a given number. It represents the positive value that satisfies the equation $x^2 = a$, where $a$ is the number whose square root is being calculated.

5 Must Know Facts For Your Next Test

1. Square roots are essential in solving equations in quadratic form, which are equations that can be rearranged into the form $ax^2 + bx + c = 0$.
2. The square root of a negative number is an imaginary number, which is not a real number and cannot be represented on the number line.
3. The square root of a perfect square can be simplified, while the square root of a non-perfect square is an irrational number.
4. Square roots can be used to find the lengths of the sides of a right triangle, as described by the Pythagorean Theorem.
5. Calculators and computers use algorithms to approximate the square root of a number to a high degree of accuracy.

Review Questions

• Explain how square roots are used to solve equations in quadratic form.
  • Equations in quadratic form can be solved by isolating the variable and then taking the square root of both sides of the equation. This allows the equation to be rewritten in the form $x = \pm \sqrt{b^2 - 4ac}$, where $a$, $b$, and $c$ are the coefficients of the quadratic equation. The square root operation is essential in this process, as it provides the two possible solutions to the equation.
• Describe the relationship between square roots and perfect squares.
  • Perfect squares are numbers that can be expressed as the product of two equal integers, such as 4, 9, and 16. The square root of a perfect square is the integer that, when multiplied by itself, yields the original number. For example, the square root of 9 is 3, because $3 \times 3 = 9$. In contrast, the square root of a non-perfect square, such as 2 or 3, is an irrational number that cannot be expressed as a simple fraction.
• Analyze the significance of square roots in the context of the Pythagorean Theorem.
  • The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides. This relationship can be expressed as the equation $a^2 + b^2 = c^2$, where $a$ and $b$ are the lengths of the shorter sides, and $c$ is the length of the hypotenuse. To find the length of the hypotenuse, we can take the square root of the sum of the squares of the other two sides, which is a crucial application of square roots in geometry.

Related terms

Radical Symbol: The symbol $\sqrt{}$ used to denote a square root operation.

Perfect Square: A number that can be expressed as the product of two equal integers, such as 4, 9, 16, etc.

Irrational Number: A number that cannot be expressed as a simple fraction, such as $\sqrt{2}$, $\sqrt{3}$, and $\pi$.