Formation of the Quadratic Equation whose Roots are Given

We will learn the formation of the quadratic equation whose roots are given.

To form a quadratic equation, let α and β be the two roots.

Let us assume that the required equation be ax2 + bx + c = 0 (a ≠ 0).

According to the problem, roots of this equation are α and β.

Therefore,

α + β = - ba and αβ = ca.

Now, ax2 + bx + c = 0

⇒ x2 + bax + ca = 0 (Since, a ≠ 0)

⇒ x2 - (α + β)x + αβ = 0, [Since, α + β = -ba and αβ = ca]

⇒ x2 - (sum of the roots)x + product of the roots = 0

⇒ x2 - Sx + P = 0, where S = sum of the roots and P = product of the roots ............... (i)

Formula (i) is used for the formation of a quadratic equation when its roots are given.

For example suppose we are to form the quadratic equation whose roots are 5 and (-2). By formula (i) we get the required equation as

x2 - [5 + (-2)]x + 5 (-2) = 0

⇒ x2 - [3]x + (-10) = 0

⇒ x2 - 3x - 10 = 0


Solved examples to form the quadratic equation whose roots are given:

1. Form an equation whose roots are 2, and - 12.

Solution:

The given roots are 2 and -12.

Therefore, sum of the roots, S = 2 + (-12) = 32

And tghe product of the given roots, P = 2 -12 = - 1.

Therefore, the required equation is x2 – Sx + p

i.e., x2 - (sum of the roots)x + product of the roots = 0

i.e., x2 - 32x – 1 = 0

i.e, 2x2 - 3x - 2 = 0


2. Find the quadratic equation with rational coefficients which has 13+22 as a root.

Solution:

According to the problem, coefficients of the required quadratic equation are rational and its one root is 13+22 = 13+22322322 = 32298 = 3 - 2√2.

We know in a quadratic with rational coefficients irrational roots occur in conjugate pairs).

Since equation has rational coefficients, the other root is 3 + 2√2.

Now, the sum of the roots of the given equation S = (3 - 2√2) + (3 + 2√2) = 6

Product of the roots, P = (3 - 2√2)(3 + 2√2) = 32 - (2√2)2 = 9 - 8 = 1

Hence, the required equation is x2 - Sx + P = 0 i.e., x2 - 6x + 1 = 0.


2. Find the quadratic equation with real coefficients which has -2 + i as a root (i = √-1).

Solution:

According to the problem, coefficients of the required quadratic equation are real and its one root is -2 + i.

We know in a quadratic with real coefficients imaginary roots occur in conjugate pairs).

Since equation has rational coefficients, the other root is -2 - i

Now, the sum of the roots of the given equation S = (-2 + i) + (-2 - i) = -4

Product of the roots, P = (-2 + i)(-2 - i) = (-2)2 - i2 = 4 - (-1) = 4 + 1 = 5

Hence, the required equation is x2 - Sx + P = 0 i.e., x2 - 4x + 5 = 0.





11 and 12 Grade Math 

From Formation of the Quadratic Equation whose Roots are Given to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. 5th Grade Pie Chart | Definition of Pie Chart | Construction |Examples

    Jul 31, 25 05:12 PM

    Pie Chart Circle
    Data can also be represented in a circle. This method, to represent data, is called a pie chart. Let us understand this method with the help of an example.

    Read More

  2. Frequency Distribution |Tally Marks |Frequency Distribution Table

    Jul 31, 25 12:23 PM

    Frequency Table
    What is frequency distribution?The number of times a particular observation occurs in a given data is called its frequency. In 7ᵗʰ grade and 8ᵗʰ grade frequency distribution,

    Read More

  3. 5th Grade Bar Graph | Definition | Interpret Bar Graphs|Free Worksheet

    Jul 31, 25 05:16 AM

    Draw a Vertical Bar Graph
    We learn how to represent the data on the bar graph. Data can be represented by bars (like rectangle) whose lengths represent numerical values. One can use horizontal or vertical bars. Instead of rect…

    Read More

  4. Construction of Bar Graphs | Examples on Construction of Column Graph

    Jul 31, 25 03:35 AM

    What is Bar Graph?
    Now we will discuss about the construction of bar graphs or column graph. In brief let us recall about, what is bar graph? Bar graph is the simplest way to represent a data. In consists of rectangular…

    Read More

  5. Successor and Predecessor | Successor of a Whole Number | Predecessor

    Jul 29, 25 12:59 AM

    Successor and Predecessor
    The number that comes just before a number is called the predecessor. So, the predecessor of a given number is 1 less than the given number. Successor of a given number is 1 more than the given number…

    Read More