Introduction
In algebra, a quadratic equation is any equation that can be rearranged in standard form y = ax²+ bx + c, where x represents an unknown value, and a, b, and c represent known numbers, where a ≠ 0. If b = 0, the equation is known as the pure quadratic equation and if b ≠ 0, then the equation is known as the adfected quadratic
Roots of a Quadratic Equation
The two values of x which are obtained by solving the quadratic equation is known as the roots of a quadratic equation. These roots of the quadratic equation are also called the zeros of the equation. For example, the roots of the equation x² – 3x – 4 = 0 are x = -1 and x = 4 because each of them satisfies the equation. i.e.
- At x = -1, (-1)² – 3(-1) – 4 = 1 + 3 – 4 = 0
- At x = 4, (4)² – 3(4) – 4 = 16 – 12 – 4 = 0
There are various methods to find the roots of any quadratic equation and the usage of the quadratic formula is one of them.
Nature of Roots of the Quadratic Equation
In general, the roots of a quadratic equation are usually represented to by alpha (α), and beta (β). Here we shall learn more about how to find the nature of roots of a quadratic equation without actually finding the roots of the equation. The nature of roots of a quadratic equation can be found without actually finding the roots (α, β) of the given equation and it is possible by taking the discriminant value, which is part of the formula to solve the quadratic equation.
Discriminant: D = b² – 4ac
- D > 0, the roots are real and distinct
- D = 0, the roots are real and equal.
- D < 0, the roots do not exist or the roots are imaginary.
Exercise – 6.1
Introduction
In algebra, a quadratic equation is any equation that can be rearranged in standard form y = ax²+ bx + c, where x represents an unknown value, and a, b, and c represent known numbers, where a ≠ 0. If b = 0, the equation is known as the pure quadratic equation and if b ≠ 0, then the equation is known as the adfected quadratic
Symmetric Function of Roots
Symmetric functions of the roots of a quadratic equation are those functions in which the two roots are so involved that the function is unaltered when the two roots are interchanged. For instance, the sum α+β and the product αβ of the roots, we have considered in the last two sections, are symmetric functions of α and β.
Sum and Product of Roots of Quadratic Equation
The coefficient of x², x term, and the constant term of the quadratic equation ax²+ bx + c = 0 are very useful in determining the sum and product of the roots of the quadratic equation. It can be directly calculated from the equation, without actually finding the roots of the quadratic equation. For a quadratic equation ax² + bx + c = 0, the sum and product of the roots are as follows.
- Sum of the Roots: α + β = -b/a = – Coefficient of x/ Coefficient of x²
- Product of the Roots: αβ = c/a = Constant term/ Coefficient of x²
Exercise – 6.2
ntroduction
In algebra, a quadratic equation is any equation that can be rearranged in standard form y = ax²+ bx + c, where x represents an unknown value, and a, b, and c represent known numbers, where a ≠ 0. If b = 0, the equation is known as the pure quadratic equation and if b ≠ 0, then the equation is known as the adfected quadratic
Roots of a Quadratic Equation
The two values of x which are obtained by solving the quadratic equation is known as the roots of a quadratic equation. These roots of the quadratic equation are also called the zeros of the equation. For example, the roots of the equation x² – 3x – 4 = 0 are x = -1 and x = 4 because each of them satisfies the equation. i.e.
- At x = -1, (-1)² – 3(-1) – 4 = 1 + 3 – 4 = 0
- At x = 4, (4)² – 3(4) – 4 = 16 – 12 – 4 = 0
There are various methods to find the roots of any quadratic equation and the usage of the quadratic formula is one of them.
Common Roots
In this section, we shall obtain conditions under which two given quadratic equations may have one root common or both roots common.
a) One Root Common
Let ax²+ bx + c = 0 and a’x² + b’x + c = 0 be two equations. Suppose that α is a root common to both the equations. Then
aα² + bα + c = 0
a’α² + b’α + c = 0
By the rule of cross-multiplication, we get
α²/bc’ – b’c = α/ca’ – c’a = 1/ab’ – a’b
This gives α = (bc’ – b’c)/(ca’ – c’a) and also α = (ca’ – c’a)/(ab’ – a’b)
Combining the two, we have the required condition
(bc’ – b’c) (ab’ – a’b) = (ca’ – c’a)²
and the common root is
(bc – b’c) / (ca’ – c’a) Or (ca’ – c’a) / (ab’ – a’b)
b) Two Roots Common
If the quadratic equations have both roots common and if α and β be the common roots, then
α + β = -b/a = -b’/a Or a/a’ = b/b’
and
αβ = c/a = c’/a’ Or a/a’ = c/c’
Hence, a/a’ = b/b’ = c/c’,
which is the required condition for the equations to have both roots common.
