CBSE Class 10t Chapter 4 Quadratic Equations Details & Preparations Downloads

As we step into the realm of quadratic equations, the intricacies of this mathematical chapter unfold. In this blog post, we'll delve into the definition, properties, and various methods of solving quadratic equations, unraveling the beauty and importance of this fundamental concept in mathematics.

Quadratic Equations Demystified Navigating the Complexity of a Fundamental Chapter

Defining Quadratic Equations

Quadratic equations are the polynomial equations of degree 2 in one variable of type f(x) = ax2 + bx + c = 0 where a, b, c, ∈ R and a ≠ 0. It is the general form of a quadratic equation where 'a' is called the leading coefficient and 'c' is called the absolute term of f (x).

Understanding the Quadratic Formula

The quadratic formula helps us solve any quadratic equation. First, we bring the equation to the form ax²+bx+c=0, where a, b, and c are coefficients. Then, we plug these coefficients in the formula: (-b±√(b²-4ac))/(2a) . See examples of using the formula to solve a variety of equations.

Quadratic equations

Quadratic equations are the polynomial equations of degree 2 in one variable of type f(x) = ax2 + bx + c = 0 where a, b, c, ∈ R and a ≠ 0. It is the general form of a quadratic equation where ‘a’ is called the leading coefficient and ‘c’ is called the absolute term of f (x). The values of x satisfying the quadratic equation are the roots of the quadratic equation (α, β).

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Graphical Representation

Visualizing quadratic equations on a coordinate plane provides a geometric understanding of parabolas, the graphical representation of quadratic functions. This exploration allows us to interpret key features such as vertex, axis of symmetry, and direction of opening.

Quadratic Equation Definition

A quadratic polynomial, when equated to zero, becomes a quadratic equation. In other terms, a quadratic equation is a second degree algebraic equation. The values of x satisfying the equation are called the roots of the quadratic equation.

General from: ax² + bx + c = 0

Here, a, b, c, ∈ R and a ≠ 0

Examples: 3x² + x + 5 = 0, -x² + 7x + 5 = 0, x² + x = 0.

Quadratic Equation Formula

The solution or roots of a quadratic equation are given by the quadratic formula:

x = (α, β) = [-b ± √(b² – 4ac)]/2a

Roots of Quadratic Equation

The values of variables satisfying the given quadratic equation are called their roots. In other words, x = α is a root of the quadratic equation f(x), if f(α) = 0.

The real roots of equation f(x) = 0 are the x-coordinates of the points where the curve y = f(x) intersects the x-axis.

• One of the roots of the quadratic equation is zero, and the other is -b/a if c = 0
• Both the roots are zero if b = c = 0
• The roots are reciprocal to each other if a = c

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Quadratic Equations Having Common Roots

Let β be the common root (solution) of quadratic equations a1x2 + b1x + c1 = 0 and a2x2 + b2x + c2 = 0. This implies that a1β2 + b1β + c1 = 0 and a2β2 + b2β + c2 = 0.

Now, solving for β2 and β we will get:

β2/(b1c2 – b2c1) = -β/(a1c2 – a2c1) = 1/(a1b2 – a2b1) [using determinant method]

Therefore, β2 = (b1c2 – b2c1)/ (a1b2 – a2b1) . . . . . . . . . . . . . . . . (1)

And, β = (a2c1 – a1c2)/(a1b2 – a2b1) . . . . . . . . . . . . . . . . (2)

On squaring equation (2) and equating it with equation (1), we get:

(a1b2 – a2b1)/(b1c2 – b2c1) = (a2c1 – a1c2)2

Hence, it is the required condition for quadratic equations having one common root.

If both the roots of quadratic equations a1x2 + b1x + c1 = 0 and a2x2 + b2x + c2 = 0 are common then:

a1/a2 = b1/b2 = c1/c2

If α is a repeated root, i.e., the two roots are α, α of equation f(x) = 0, then α will be a root of the derived equation.

f’(x) = 0 where f’(x) = df/dx

If α is a repeated root common in f(x) = 0 and ϕ(x) = 0, then α is a common root both in f’(x) = 0 and ϕ ‘(x) = 0.

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FAQ

Q1. What are the roots of a quadratic equation?

Ans  The roots are the values of the variable that make the quadratic equation equal to zero.

Q2. How can I find the roots of a quadratic equation?

Ans The roots can be found using the quadratic formula or by factoring the equation.

Q3. Can a quadratic equation have only one root?

Ans Yes, if the discriminant (the part under the square root in the quadratic formula) is zero, the equation has a single repeated root.

Q4. What is the discriminant, and how does it determine the nature of roots?

Ans The discriminant is the expression inside the square root in the quadratic formula It determines whether the roots are real, equal, or imaginary.

Q5. How does factoring help find the roots of a quadratic equation?

Ans Factoring involves expressing the quadratic equation as a product of two linear factors, making it easier to identify the values that satisfy the equation.