Values of a and b are –2 and 1, respectively, if a and b are Roots of Equation x2 + bx + a = 0, and a, b ≠ 0, then find Values of a and b.
Now let’s find the solution for the same.
If a and b are Roots of Equation x2 + bx + a = 0, and a, b ≠ 0, then find Values of a and b.
Solution:
Given equation,
- 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
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,
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,
Values of a and b are –2 and 1, respectively.
Practice Problems
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 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, 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.
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, 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
Roots of new equation are 1/α and 1/β.
Sum of New Roots = 1/α + 1/β
= (α + β)/αβ = (–10/3)/(8/3)
= –10/8
Product of New Roots = 1/αβ = 1/(8/3) = 3/8
So, 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
Write the standard form of Quadratic equations.
Standard form of the quadratic equation is ax2 + bx + c = 0, where a, b are coefficient of variables and c is the constant.
Write the uses of Quadratic equations.
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.
What do you mean by quadratic equation?
Quadratic equation is a second-degree equation of the form,
ax2 + bx + c = 0
What are the Number of Roots of the Quadratic Equation?
Quadratic equation is a second-degree equation so it has two roots or two solutions.
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