Quadratic Equations:
The chapter 'Quadratic Equations' carrying the weightage of \(6\) marks in the board examination. It covers the concepts like Framing Quadratic Equation using given situation/case study, Solving Quadratic Equation by Factorization, Solving Quadratic Equation by Quadratic Formula and Nature of Roots.
Most possible variation of question types that we can expect in board exam as per previous year question paper are discussed below.
| Total Marks \(5\) to \(6\) | |
| Variation \(1\) | Variation \(2\) |
Total Mark \(= 6\)
|
Total Mark \(= 5\)
|
To prepare well for the board examination, it is necessary to understand the following concepts clearly.
- Quadratic Equation - Check whether the given equation is quadratic or not, Framing quadratic equation for the given situation
- Solving Quadratic Equation - Factorisation Method, Quadratic Formula Method
- Nature of roots - Discrimnants.
| Important Concept (Learning Outcomes) | Expected Question Type | Concept dealt with |
|
Quadratic Equations
|
Sec D
|
|
|
Solving Quadratic Equation
|
Sec A, Sec D, Sec E | |
|
Nature of roots
|
Sec A, Sec D |
Let us recall the concepts in Polynomial:
1. A quadratic equation in the variable \(x\) is of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), \(c\) are real numbers and \(a\neq 0\). A real number \(p\) is said to be a root of the quadratic equation \(ax^2+ bx + c = 0\), if \(ap^2+ bp+c = 0\). The zeroes of the quadratic polynomial \(ax^2+ bx + c\) and the roots of the quadratic equation \(ax^2+ bx + c = 0\) are the same.
2. Solving the quadratic equation can be done in two different way:
- Solving by factorization.
- Solving by quadratic formula.
Solving by Factorization: If we can factorise \(ax^2+ bx + c\), \(a \neq 0\), into a product of two linear factors, then the roots of the quadratic equation \(ax^2+ bx + c = 0\) can be found by equating each factor to zero.
Solving by Quadratic Formula: The formula for finding roots of the quadratic roots
\(ax^2+bx+c=0\) is \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\)
3. Nature of Roots: A quadratic equation \(ax^2+ bx + c = 0\) has
| Discrimnent | Type of Roots |
| \(b^2-4ac > 0\) | Real and Distinct |
|
\(b^2-4ac = 0\)
|
Real and Equal |
| \(b^2-4ac < 0\) | No Real Roots |
Common mistakes to avoid:
1. Without converting standard form start factoring or applying the quadratic forula without rearranging the terms.
2. While checking for quadratic equation, without expanding all the terms, don't assume an equation is quadratic or not.
3. Verify middle-term spliting signs: Ensure the chosen factors multiply to equal \(ac\) and add up to equal \(b\) with exact signs.
4. Include both positive and negaive roots: takisn square roots requires both positive and negative roots.
5. Solving word problems:
Reject invalid real-world answers: Speed, distance, age and time cannot be negative. Always reject negative roots explicitly with a brief reason.
Not defining the variable clearly before forming the quadratic equation.
6. Incorrectly interpreting the discriminant values.
Exam Tips & Tricks:
1. Always convert to standard form before identifying the coefficients.
2. Factorisation: For \(x^2 + bx + c\), find two numbers whose sums equal the value of \(b\) and the product equals \(c\).
3. Use the quickest method:
- Easily factorisable \(\Rightarrow \text{Factorisation method}\)
- Not easily factorisable \(\Rightarrow \text{Quadratic formula}\)
4. Word problems:
- In word problems, define the variable first and from the equation carefully before solving.
- Variable assignment: First identify the independent and dependent quantities. Always assign the variable to the independent quantity and express the dependent quantity in terms of it.
- Check the meaning of the roots: In applications involving length, age, distance etc., reject values that do not satisfy the given conditions.