square root

(noun)

A root of degree 2, written in the form a\sqrt{a}√​a​​​.

Related Terms

  • cube root
  • radical expression
  • root

Examples of square root in the following topics:

  • Radical Functions

    • An expression with roots is called a radical function, there are many kinds of roots, square root and cube root being the most common.
    • If fourth root of 2401 is 7, and the square root of 2401 is 49, then what is the third root of 2401?
    • If the square root of a number is taken, the result is a number which when squared gives the first number.
    • Roots do not have to be square.
    • However, using a calculator can approximate the square root of a non-square number:3=1.73205080757\sqrt {3} = 1.73205080757√​3​​​=1.73205080757
  • Imaginary Numbers

    • There is no such value such that when squared it results in a negative value; we therefore classify roots of negative numbers as "imaginary."
    • A radical expression represents the root of a given quantity.
    • When the radicand (the value under the radical sign) is negative, the root of that value is said to be an imaginary number.
    • Specifically, the imaginary number, iii, is defined as the square root of -1: thus, i=−1i=\sqrt{-1}i=√​−1​​​.
    • We can write the square root of any negative number in terms of iii.
  • Radical Equations

    • When solving equations that involve radicals, begin by asking: is there an x under the square root?
    • If there is an xxx, or variable, under the square root, the problem must be approached differently.
    • Square both sides of the equation if the radical is a square root; Cube both sides if the radical is a cube root.
    • To undo the radical symbol (square root), square both sides of the equation.
    • (6x−2)2=(10)2(\sqrt{6x-2})^2=(10)^2(√​6x−2​​​)​2​​=(10)​2​​,  squaring a square root leaves the expression under the square root symbol and 101010 squared is 100100100
  • Solving Problems with Radicals

    • Roots are written using a radical sign, and a number denoting which root to solve for.
    • When none is given, it is an implied square root.
    • Roots are written using a radical sign.
    • If there is no denotation, it is implied that you are finding the square root.
    • Otherwise, a number will appear denoting which root to solve for.
  • The Discriminant

    • The discriminant Δ=b2−4ac\Delta =b^2-4acΔ=b​2​​−4ac is the portion of the quadratic formula under the square root.
    • If Δ{\Delta}Δ is positive, the square root in the quadratic formula is positive, and the solutions do not involve imaginary numbers.
    • If Δ{\Delta}Δ is equal to zero, the square root in the quadratic formula is zero:
    • If Δ{\Delta}Δ is less than zero, the value under the square root in the quadratic formula is negative:
    • This means the square root itself is an imaginary number, so the roots of the quadratic function are distinct and not real.
  • Introduction to Radicals

    • Roots are the inverse operation of exponentiation.
    • For example, the following is a radical expression that reverses the above solution, working backwards from 49 to its square root:
    • For now, it is important simplify to recognize the relationship between roots and exponents: if a root rrr is defined as the nthn \text{th}nth root of xxx, it is represented as
    • If the square root of a number xxx is calculated, the result is a number that when squared (i.e., when raised to an exponent of 2) gives the original number xxx.
    • This is read as "the square root of 36" or "radical 36."
  • Simplifying Radical Expressions

    • where nnn is the degree of the root.
    • A root of degree 2 is called a square root and a root of degree 3, a cube root.
    • Roots of higher degrees are referred to using ordinal numbers, as in fourth root, twentieth root, etc.
    • First, look for a perfect square under the square root sign and remove it:
    • First, notice that there is a perfect square under the square root symbol, and pull that out: 16x7x24=4⋅x7x24=4⋅x7x24\frac{\sqrt{16}\sqrt{x^7}}{\sqrt[4]{x^2}} = \frac{4 \cdot \sqrt{x^7}}{\sqrt[4]{x^2}} = 4 \cdot \frac{\sqrt{x^7}}{\sqrt[4]{x^2}}​​4​​√​x​2​​​​​​​√​16​​​√​x​7​​​​​​​=​​4​​√​x​2​​​​​​​4⋅√​x​7​​​​​​​=4⋅​​4​​√​x​2​​​​​​​√​x​7​​​​​​​
  • Factoring Perfect Square Trinomials

    • When a trinomial is a perfect square, it can be factored into two equal binomials.
    • Note that if a binomial of the form a+ba+ba+b is squared, the result has the following form: (a+b)2=(a+b)(a+b)=a2+2ab+b2.(a+b)^2=(a+b)(a+b)=a^2+2ab+b^2.(a+b)​2​​=(a+b)(a+b)=a​2​​+2ab+b​2​​. So both the first and last term are squares, and the middle term has factors of 2,2, 2, aaa, and b,b,b, where the latter are the square roots of the first and last term respectively.
    • For example, if the expression 2x+32x+32x+3 were squared, we would obtain (2x+3)(2x+3)=4x2+12x+9.(2x+3)(2x+3)=4x^2+12x+9.(2x+3)(2x+3)=4x​2​​+12x+9. Note that the first term 4x24x^24x​2​​ is the square of 2x2x2x while the last term 999 is the square of 333, while the middle term is twice 2x⋅32x\cdot32x⋅3.
    • Suppose you were trying to factor x2+8x+16.x^2+8x+16.x​2​​+8x+16. One can see that the first term is the square of xxx while the last term is the square of 444.
    • Since the first term is 3x3x3x squared, the last term is one squared, and the middle term is twice 3x⋅13x\cdot 13x⋅1, this is a perfect square, and we can write:
  • Completing the Square

    • The method of completing the square allows for the conversion to the form:
    • Once completing the square has been performed, the quadratic is easy to solve; because there is only one place where the variable xxx is squared, the (x−h)2(x-h)^2(x−h)​2​​ term can be isolated on one side of the equation, and then the square root of both sides can be taken.
    • This quadratic is not a perfect square.  
    • The closest perfect square is the square of 555, which was determined by dividing the bbb term (in this case 101010) by two and producing the square of the result.
    • Solve for the zeros of a quadratic function by completing the square
  • Factoring a Difference of Squares

    • When a quadratic is a difference of squares, there is a helpful formula for factoring it.
    • Taking the square root of both sides of the equation gives the answer x=±ax = \pm ax=±a.
    • Using the difference of squares is just another way to think about solving the equation.
    • If you recognize the first term as the square of xxx and the term after the minus sign as the square of 444, you can then factor the expression as:
    • If we recognize the first term as the square of 4x24x^24x​2​​ and the term after the minus sign as the square of 333, we can rewrite the equation as:
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