square

Examples of square in the following topics:

• 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 $a+b$ is squared, the result has the following form: $(a+b)^2=(a+b)(a+b)=a^2+2ab+b^2.$ So both the first and last term are squares, and the middle term has factors of $2, $ $a$, and $b,$ where the latter are the square roots of the first and last term respectively.
  • For example, if the expression $2x+3$ were squared, we would obtain $(2x+3)(2x+3)=4x^2+12x+9.$ Note that the first term $4x^2$ is the square of $2x$ while the last term $9$ is the square of $3$, while the middle term is twice $2x\cdot3$.
  • Suppose you were trying to factor $x^2+8x+16.$ One can see that the first term is the square of $x$ while the last term is the square of $4$.
  • Since the first term is $3x$ squared, the last term is one squared, and the middle term is twice $3x\cdot 1$, 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 $x$ is squared, the $(x-h)^2$ 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 $5$, which was determined by dividing the $b$ term (in this case $10$) 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 $x = \pm a$.
  • 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 $x$ and the term after the minus sign as the square of $4$, you can then factor the expression as:
  • If we recognize the first term as the square of $4x^2$ and the term after the minus sign as the square of $3$, we can rewrite the equation as:

• Standard Form and Completing the Square

  • The standard form of a quadratic equation is useful for completing the square, which is used to graph the equation.
  • Completing the square is a technique for converting a quadratic polynomial of the form:
  • Completing the square may be used to solve any quadratic equation.

• Radical Functions

  • 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.
  • If a root of a whole number is squared root, which is not itself the square of a rational number, the answer will have an infinite number of decimal places.
  • However, using a calculator can approximate the square root of a non-square number:$\sqrt {3} = 1.73205080757$
  • Writing the square root of 3 or any other non-square number as $\sqrt {3}$ is the simplest way to represent the exact value.

• Radical Equations

  • When solving equations that involve radicals, begin by asking: is there an $x$ under the square root?
  • In this case, both sides must be squared to get rid of the radical.
  • Now, to undo the radical symbol (square root), square both sides of the equation (recall that squaring a square root removes the radical):
  • This is incorrect, because the square root is defined to be only the positive root, $10$.
  • Solve a radical equation by squaring both sides of the equation

• 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."
  • Specifically, the imaginary number, $i$, is defined as the square root of -1: thus, $i=\sqrt{-1}$.
  • We can write the square root of any negative number in terms of $i$.

• Determinants of 2-by-2 Square Matrices

  • The determinant of a $2\times 2$ square matrix is a mathematical construct used in problem solving that is found by a special formula.
  • In linear algebra, the determinant is a value associated with a square matrix.
  • Explain what a determinant represents, how to find one, and why only square matrices have them

• Graphing Quadratic Equations in Vertex Form

  • If you want to convert a quadratic in vertex form to one in standard form, simply multiply out the square and combine like terms.
  • The process is called "completing the square."  
  • Then we square that number.
  • Thus for this example, we divide $4$ by $2$ to obtain $2$ and then square it to obtain $4$.
  • We then complete the square within the parentheses.

• Finding Factors of Polynomials

  • The second type of factoring is based on the "squaring" formulae:
  • This is because if a negative is squared, the answer is positive.
  • This formula can be run in reverse whenever subtracting two perfect squares.
  • Note also that when we are working with real numbers, all positive numbers are squares.
  • It is important to note that the sum of two squares cannot be factored.