If a and b are the roots of the equation x² + bx + a = 0, and a, b ≠ 0, then find the values of a and b

Algebra is a branch of mathematics that deals with unknown variables and the mathematical operations performed on them. An algebraic expression is a mathematical statement that is a combination of terms by the basic arithmetic operations such as addition, subtraction, multiplication, division, etc. a, b, m, n, x, y, z, etc are some examples of variables where a variable is a symbol that doesn’t have any fixed value. A constant is a symbol having a fixed numerical value. Some examples of constants are 1, 6, (2/9), -(5/6), √7, etc. A term might be a variable only (or) a constant only (or) both variables and constants that are combined by multiplication or division. 4x, -(2x²), (m/3), √(2x/3), etc. are some examples of terms. A coefficient is a value that is placed before and multiplied by a variable. When a polynomial is expressed in its standard form, the degree is the highest integral power of the variables of its terms.

Quadratic Equation

A quadratic equation is referred to as an algebraic equation whose highest degree is two, with one unknown variable and it has two roots. The general form of a quadratic equation in x is written as 

ax² + bx + c = 0, 

Where a ≠ 0, a, b, & c ∈ R

In the equation given above, x is the unknown variable, a and b are the coefficients of x² and x, respectively, and c is a constant.

The values of the unknown variable x that satisfy the equation are the solutions to the given quadratic equation. These solutions are known as roots or zeros of the quadratic equation. A quadratic equation always has two solutions as the highest degree of the variable is two. The roots, or zeros, of a quadratic equation, are the values of “x” that satisfy the quadratic equation, i.e. when we substitute the values of x on the left-hand side of the equation, it will equal zero.

The formula for finding the roots of a quadratic equation, ax² + bx + c = 0, is given as follows:

x = [–b ± √(b² – 4ac)]/2a, a ≠ 0

Here, the sign of plus or minus in the formula indicates that there will be two solutions for x.

Sum and Product of Roots of Quadratic Equation

If α and β be the roots of the quadratic equation ax² + bx + c = 0, then

1. The sum of roots (α + β) = –b/a = – coefficient of x/ coefficient of x²
2. Product of roots (αβ) = c/a = constant/coefficient of x²

We can write the quadratic equation for the given roots α and β as follows:

x² – (α + β)x + αβ = 0

If a and b are the roots of the equation x² + bx + a = 0, and a, b ≠ 0, then find the values of a and b.

Solution:

Given equation is x² + bx + a = 0, a ≠ 0, b ≠ 0, a, b ∈ R.

a and b are the roots of the given equation.

We know that, for a quadratic equation ax² + bx + c = 0

The sum of roots = –b/a = – coefficient of x/coefficient of x²

Product of roots = c/a = constant/coefficient of x²

So, for the given quadratic equation x² + bx + a = 0,

The sum of roots = –b/1 

⇒ a + b = – b

⇒ a + 2b = 0   ————— (1)

Product of roots = c/a = a/1 

⇒ ab = a

⇒ ab – a= 0

⇒ (b – 1)a= 0

So, (b – 1) = 0 {Since, a ≠ 0}

⇒ b = 1

Now, substituting the value of a= 1 in equation (1), we get

⇒ a + 2(1) = 0

⇒ a = –2

Hence,

The values of a and b are  –2 and 1, respectively.

Practice Problems based on Quadratic equations

Problem 1: Find the sum and product of roots of the quadratic equation 5x² + 7x + 2 = 0.

Solution:

Given equation: 5x² + 7x + 2 = 0

By comparing the given equation with the standard equation ax² + bx + c = 0,

we get have a = 5, b= 7 and c = 2

Now, substitute the values in the formulae of sum and product of roots.

Sum of roots = -b/a = –7/5

Product of roots = c/a = 2/5

Problem 2: What is the quadratic equation if the sum of its roots is –5 and the products of its roots is 6?

Solution:

Given data,

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

Sum of roots (α + β) = –5

Product of roots (αβ) = 6

Now, the required equation is x² – (α + β)x + αβ = 0

⇒ x² – (–5)x + 6 = 0

⇒ x² + 5x + 6 = 0

Thus, x² + 5x + 6 = 0 is the required quadratic equation.

Problem 3: Find the quadratic equation whose roots are 6 and 7, respectively.

Solution:

Given,

6 and 7 are roots of a quadratic equation i.e.,

α = 5, and β = 8.

The quadratic equation whose roots are α and β, is x² – (α + β)x + αβ = 0.

⇒ x² – (6 + 7)x + (6 × 7) = 0

⇒ x² –13x + 42 = 0

Hence, the required quadratic equation is x² – 13x + 42 = 0.

Problem 4: If α and β are the roots of the quadratic equation 3x² + 10x + 8 = 0, then find the quadratic equation whose roots are 1/α and 1/β.

Solution:

Given equation: 3x² + 10x + 8 = 0

Sum of roots (α + β) = –10/3 

Product of roots (αβ) =  8/3

The roots of the new equation are 1/α and 1/β.

Their sum = 1/α + 1/β 

= (α + β)/αβ = (–10/3)/(8/3)

= –10/8

Their product = 1/αβ = 1/(8/3) = 3/8

So, the required equation is,

⇒ x² – (1/α + 1/β)x + 1/αβ = 0

⇒ x² – (–10/8)x + 3/8 = 0

Multiplying 8 on both sides, we get

⇒ 8x² + 10x + 3 = 0

Thus, 8x² + 10x + 3 = 0 is the required quadratic equation.

Problem 5: Find the roots of the equation: 2x² + x – 3 = 0.

Solution:

Given equation: 2x² + x – 3 = 0.

By comparing the given equation with the standard equation ax² + bx + c = 0,

we get have a = 2, b = 1 and c = –3

Now, substitute the values in the formula for finding the roots of a quadratic equation

x = [–b ± √(b² – 4ac)]/2a

⇒ x = [–1 ± √(1² – 4(2)(–3) ]/2(2)

⇒ x = [–1 ± √(1+24)]/4

⇒ x = [–1 ± √25]/4 = [–1 ± 5]/4

⇒ x = (–1–5)/4 = –6/4 = –3/2

(or)

⇒ x = (–1+5)/4 = 4/4 = 1

So, the roots of the given quadratic equation are –3/2 and 1.

Frequently Asked Questions

Question 1: Write the standard form of Quadratic equations.

Answer:

The standard form of the quadratic equation is ax² + bx + c = 0, where a, b are coefficient of variables and c is the constant.

Question 2: Write the uses of Quadratic equations.

Answer: 

Uses of quadratic equations are, they are used in everyday scenarios for a variety of reasons, like finding the areas of various object, calculating various quantities etc.

Question 3: What do you mean by quadratic equation?

Answer: 

Quadratic equation is a second-degree equation of the form 

ax² + bx + c = 0. 

Where a, b, are the coefficients of variable term, c is the constant term, and x is the variable.

Question 4: What are the Number of Roots of the Quadratic Equation?

Answer:

Quadratic equation is a second-degree equation so it has two roots or two solutions.