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    • Quadratic Equations

      Introduction

      A polynomial of degree 2 is called quadratic polynomial. The name Quadratic comes from "quad" means square, because the variable gets squared (like x2).

       

      The general form of a quadratic polynomials is ax2 + bx + c, where a, b, c are real numbers such that a ≠ 0 and x is a variable. When we equate this polynomial to zero, we get a quadratic equation i.e. ax2 + bx + c = 0 and when we write the terms of p(x) in descending order of their degrees, then we get the "Standard Form" of a Quadratic Equation i.e. ax2 + bx + c = 0. We can also denote a quadratic polynomial ax2 + bx + c by p(x) i.e. p(x) = ax2 + bx + c. It is also called an "Equation of degree 2" (because of the power "2" on the x).

       

      The real numbers a, b and c are known as the coefficients and are also known as the real constants because they are fixed and do not depend on the values of variable x.

       

      Here are some more examples:



      Example 1: 2x2 -3x + 4 = 0 (Here a=2, b=-3 and c=4)

       

      Example 2: x2 + 9 = 0 (Here a=1, and we don't usually write "1x2", b=0 and c=9)

       

      Example 3: x2 = 3x – 1 (In this case we will move all terms to left hand side i.e. x2 − 3x + 1 = 0. Here a=1, b=−3, c=1)

       

      Example 4: 2(p2 – 2p) = 5 (Solve the brackets and move all terms to left hand side i.e. 2p2 − 4p − 5 = 0. Here a=2, b=−4, c=−5)

       

      Example 5: Jia and Tia together have 45 pencils. Both of them lost 5 pencils each, and the product of the number of pencils they now have is 124. Find out how many pencils each had.

      Solution: Let the number of pencils Jia had be x.

      Then the number of pencils Tia had = 45 – x (left out of total belongs to Tia)

      The number of pencils left with Jia, when she lost 5 pencils = x – 5

      The number of pencils left with Tia, when she lost 5 pencils

      = 45 – x – 5

      = 40 – x



      Value of Quadratic Polynomial

      Let’s p(x) = ax2 + bx + c be a quadratic polynomial and let ‘α’ be the real number.

      Then, aα2 + bα + c is known as the value of the quadratic polynomial p(x) and it is denoted by p (α), i.e., p(α)= aα2 + bα + c.

      Thus, the value of a quadratic polynomial ax2 + bx + c at x = α is the value of the expression ax2 + bx + c obtained by substituting x = α.

      Example1: Find the values of the quadratic polynomial p(x) = 2x2 -3x + 5 at x = -1 and x = 2.

      Solution: Here, p(x) = 2x2 -3x + 5.

      Case 1: Now substitute the value of x by -1 in the given quadratic polynomial.

      Therefore,

      p (-1) = 2 x (-1)2 – 3 x (-1) + 5

      = 2 + 3 + 5

      = 10.

      Case 2: Now substitute the value of x by 2 in the given quadratic polynomial.

      Therefore,

      p (2) = 2 x (2)2 – 3 x (2) + 5

      = 8 - 6 + 5

      = 7.



      Zero of a Quadratic Polynomial

       A real number ‘α’ is called a zero of a quadratic polynomial p(x) = ax2 + bx + c, if p (α) = 0. Thus, a zero of a quadratic polynomial is the value of the variable for which the value of the polynomial becomes zero. Every quadratic polynomial can have at most two zeros.

      Example 1: Show that 2 is a zero of the quadratic polynomial p(x) = x2 = x – 6.

      Solution: Here, p(x) = x2 = x – 6.

      Substitute value of x by 2

      Therefore, p (2) = 22 + 2 – 6

                              = 4 + 2 – 6

                              = 0

       

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