Processing math: 100%

Subscribe to our YouTube channel for the latest videos, updates, and tips.


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. Worksheet on Average | Word Problem on Average | Questions on Average

    May 19, 25 02:53 PM

    Worksheet on Average
    In worksheet on average we will solve different types of questions on the concept of average, calculating the average of the given quantities and application of average in different problems.

    Read More

  2. 8 Times Table | Multiplication Table of 8 | Read Eight Times Table

    May 18, 25 04:33 PM

    Printable eight times table
    In 8 times table we will memorize the multiplication table. Printable multiplication table is also available for the homeschoolers. 8 × 0 = 0 8 × 1 = 8 8 × 2 = 16 8 × 3 = 24 8 × 4 = 32 8 × 5 = 40

    Read More

  3. How to Find the Average in Math? | What Does Average Mean? |Definition

    May 17, 25 04:04 PM

    Average 2
    Average means a number which is between the largest and the smallest number. Average can be calculated only for similar quantities and not for dissimilar quantities.

    Read More

  4. Problems Based on Average | Word Problems |Calculating Arithmetic Mean

    May 17, 25 03:47 PM

    Here we will learn to solve the three important types of word problems based on average. The questions are mainly based on average or mean, weighted average and average speed.

    Read More

  5. Rounding Decimals | How to Round a Decimal? | Rounding off Decimal

    May 16, 25 11:13 AM

    Round off to Nearest One
    Rounding decimals are frequently used in our daily life mainly for calculating the cost of the items. In mathematics rounding off decimal is a technique used to estimate or to find the approximate

    Read More