# Relation between Roots and Coefficients of a Quadratic Equation

We will learn how to find the relation between roots and coefficients of a quadratic equation.

Let us take the quadratic equation of the general form ax^2 + bx + c = 0 where a (≠ 0) is the coefficient of x^2, b the coefficient of x and c, the constant term.

Let α and β be the roots of the equation ax^2 + bx + c = 0

Now we are going to find the relations of α and β with a, b and c.

Now ax^2 + bx + c = 0

Multiplication both sides by 4a (a ≠ 0) we get

4a^2x^2 + 4abx + 4ac = 0

(2ax)^2 + 2 * 2ax * b + b^2 – b^2 + 4ac = 0

(2ax + b)^2 = b^2 - 4ac

2ax + b = ± $$\sqrt{b^{2} - 4ac}$$

x = $$\frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$$

Therefore, the roots of (i) are $$\frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$$

Let α = $$\frac{-b + \sqrt{b^{2} - 4ac}}{2a}$$ and β = $$\frac{-b - \sqrt{b^{2} - 4ac}}{2a}$$

Therefore,

α + β = $$\frac{-b + \sqrt{b^{2} - 4ac}}{2a}$$ + $$\frac{-b - \sqrt{b^{2} - 4ac}}{2a}$$

α + β = $$\frac{-2b}{2a}$$

α + β = -$$\frac{b}{a}$$

α + β = -$$\frac{coefficient of x}{coefficient of x^{2}}$$

Again, αβ = $$\frac{-b + \sqrt{b^{2} - 4ac}}{2a}$$ × $$\frac{-b - \sqrt{b^{2} - 4ac}}{2a}$$

αβ = $$\frac{(-b)^{2} - (\sqrt{b^{2} - 4ac)}^{2}}{4a^{2}}$$

αβ = $$\frac{b^{2} - (b^{2} - 4ac)}{4a^{2}}$$

αβ = $$\frac{4ac}{4a^{2}}$$

αβ = $$\frac{c}{a}$$

αβ = $$\frac{constant term}{coefficient of x^{2}}$$

Therefore, α + β = -$$\frac{coefficient of x}{coefficient of x^{2}}$$ and αβ = $$\frac{constant term}{coefficient of x^{2}}$$ represent the required relations between roots (i.e., α and β) and coefficients (i.e., a, b and c) of equation ax^2 + bx + c = 0.

For example, if the roots of the equation 7x^2 - 4x - 8 = 0 be α and β, then

Sum of the roots = α + β = -$$\frac{coefficient of x}{coefficient of x^{2}}$$ = -$$\frac{-4}{7}$$ = $$\frac{4}{7}$$.

and

the product of the roots = αβ = $$\frac{constant term}{coefficient of x^{2}}$$ = $$\frac{-8}{7}$$ = -$$\frac{8}{7}$$.

Solved examples to find the relation between roots and coefficients of a quadratic equation:

Without solving the equation 5x^2 - 3x + 10 = 0, find the sum and the product of the roots.

Solution:

Let α and β be the roots of the given equation.

Then,

α + β = -$$\frac{-3}{5}$$ = $$\frac{3}{5}$$ and

αβ = $$\frac{10}{5}$$ = 2

To find the conditions when roots are connected by given relations

Sometimes the relation between roots of a quadratic equation is given and we are asked to find the condition i.e., relation between the coefficients a, b and c of quadratic equation. This is easily done using the formula α + β = -$$\frac{b}{a}$$ and αβ = $$\frac{c}{a}$$. This will clear when you go through illustrative examples.

1. If α and β are the roots of the equation x^2 - 4x + 2 = 0, find the value of

(i) α^2 + β^2

(ii) α^2 - β^2

(iii) α^3 + β^3

(iv $$\frac{1}{α}$$ + $$\frac{1}{ β }$$

Solution:

The given equation is x^2 - 4x + 2 = 0 ...................... (i)

According to the problem, α and β are the roots of the equation (i)

Therefore,

α + β = -$$\frac{b}{a}$$ = -$$\frac{-4}{1}$$ = 4

and αβ = $$\frac{c}{a}$$ = $$\frac{2}{1}$$ = 2

(i) Now α^2 + β^2 = (α + β)^2 - 2αβ = (4)^2 – 2 * 2 = 16 – 4 = 12.

(ii) α^2 - β^2 = (α + β)( α - β)

Now (α - β)^2 = (α + β)^2 - 4αβ = (4)^2 – 4 * 2 = 16 – 8 = 8

⇒ α - β = ± √8

⇒ α - β = ± 2√2

Therefore, α^2 - β^2 = (α + β)( α - β) = 4 * (± 2√2) = ± 8√2.

(iii) α^3 + β^3 = (α + β)^3 - 3αβ(α + β) = (4)^3 – 3 * 2 * 4 = 64 – 24 = 40.

(iv) $$\frac{1}{α}$$ + $$\frac{1}{ β }$$ = $$\frac{ α + β }{α β }$$ = $$\frac{4}{2}$$ = 2.

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.

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

## Recent Articles

1. ### Adding 1-Digit Number | Understand the Concept one Digit Number

Sep 18, 24 03:29 PM

Understand the concept of adding 1-digit number with the help of objects as well as numbers.

2. ### Addition of Numbers using Number Line | Addition Rules on Number Line

Sep 18, 24 02:47 PM

Addition of numbers using number line will help us to learn how a number line can be used for addition. Addition of numbers can be well understood with the help of the number line.

3. ### Counting Before, After and Between Numbers up to 10 | Number Counting

Sep 17, 24 01:47 AM

Counting before, after and between numbers up to 10 improves the child’s counting skills.

4. ### Worksheet on Three-digit Numbers | Write the Missing Numbers | Pattern

Sep 17, 24 12:10 AM

Practice the questions given in worksheet on three-digit numbers. The questions are based on writing the missing number in the correct order, patterns, 3-digit number in words, number names in figures…