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Complex Numbers and Quadratic Equations Class 11 Notes Maths Chapter 4
- In order to solve a quadratic equation of the form ax2 + bx + c, we first need to calculate the discriminant with the help of the formula D = b2 – 4ac.
- The solution of the quadratic equation ax2 + bx + c= 0 is given by x = [-b ± √ b2 – 4ac] / 2a
- If α and β are the roots of the quadratic equation ax2 + bx + c = 0, then we have the following results for the sum and product of roots:
α + β = -b/a
α.β = c/a
α – β = √D/a - It is not possible for a quadratic equation to have three different roots and if in any case it happens, then the equation becomes an identity.
- Nature of Roots:
Consider an equation ax2 + bx + c = 0, where a, b and c ∈ R and a ≠ 0, then we have the following cases:
1. D > 0 iff the roots are real and distinct i.e. the roots are unequal
2. D = 0 iff the roots are real and coincident i.e. equal
3. D < 0 iffthe roots are imaginary
4. The imaginary roots always occur in pairs i.e. if a+ib is one root of a quadratic equation, then the other root must be the conjugate i.e. a-ib, where a, b ∈ R and i = √-1.
Consider an equation ax2 + bx + c = 0, where a, b and c ∈Q and a ≠ 0, then
1. If D > 0 and is also a perfect square then the roots are rational and unequal.
2. If α = p + √q is a root of the equation, where ‘p’ is rational and √q is a surd, then the other root must be the conjugate of it i.e. β = p - √q and vice versa. - If the roots of the quadratic equation are known, then the quadratic equation may be constructed with the help of the formula
x2 – (Sum of roots)x + (Product of roots) = 0.
So if α and β are the roots of equation then the quadratic equation is
x2 – (α + β)x + α β = 0 - For the quadratic expression y = ax2 + bx + c, where a, b, c ∈ R and a ≠ 0, then the graph between x and y is always a parabola.
1. If a > 0, then the shape of the parabola is concave upwards
2. If a < 0, then the shape of the parabola is concave upwards - Inequalities of the form P(x)/ Q(x) > 0 can be easily solved by the method of intervals of number line rule.
- The maximum and minimum values of the expression y = ax2 + bx + c occur at the point x = -b/2a depending on whether a > 0 or a< 0.
1. y ∈[(4ac-b2) / 4a, ∞] if a > 0
2. If a < 0, then y ∈ [-∞, (4ac-b2) / 4a] - The quadratic function of the form f(x, y) = ax2+by2 + 2hxy + 2gx + 2fy + c = 0 can be resolved into two linear factors provided it satisfies the following condition: abc + 2fgh –af2 – bg2 – ch2 = 0
- In general, if α1,α2, α3, …… ,αn are the roots of the equation
f(x) = a0xn +a1xn-1 + a2xn-2 + ……. + an-1x + an, then
1.Σα1 = - a1/a0
2.Σ α1α2 = a2/a0
3.Σ α1α2α3 = - a3/a0
……… ……….
Σ α1α2α3 ……αn= (-1)n an/a0 - Every equation of nth degree has exactly n roots (n ≥1) and if it has more than n roots then the equation becomes an identity.
- If there are two real numbers ‘a’ and ‘b’ such that f(a) and f(b) are of opposite signs, then f(x) = 0 must have at least one real root between ‘a’ and ‘b’.
- Every equation f(x) = 0 of odd degree has at least one real root of a sign opposite to that of its last term.
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FAQs on Complex Numbers and Quadratic Equations Class 11 Notes Maths Chapter 4
| 1. What is a quadratic equation? |  |
Ans. A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable. It represents a curve called a parabola and can have two distinct solutions or roots.
| 2. How do you solve a quadratic equation? | |
Ans. There are several methods to solve a quadratic equation. One common method is the quadratic formula, which is x = (-b ± √(b^2 - 4ac))/(2a). Another method is factoring, where the equation is rewritten as (x - p)(x - q) = 0 and solved by finding the values of p and q. Additionally, completing the square and using the graphical method are other ways to solve quadratic equations.
| 3. What is the discriminant of a quadratic equation? | |
Ans. The discriminant of a quadratic equation is the expression inside the square root of the quadratic formula, i.e., b^2 - 4ac. It determines the nature of the solutions. If the discriminant is positive, the equation has two distinct real solutions. If it is zero, the equation has one real solution. And if it is negative, the equation has two complex solutions.
| 4. Can a quadratic equation have only one solution? | |
Ans. Yes, a quadratic equation can have only one solution. This occurs when the discriminant of the equation is zero. In this case, the parabola represented by the equation touches the x-axis at exactly one point. This type of solution is called a double root or a repeated root.
| 5. How are quadratic equations used in real life? | |
Ans. Quadratic equations have various applications in real life. They are used in physics to describe the motion of projectiles, such as the trajectory of a thrown ball. In engineering, they are used to model and analyze systems with quadratic behavior, such as the stress-strain relationship in materials. They are also used in financial mathematics to model compound interest and investment growth. Additionally, quadratic equations are used in computer graphics to create realistic curves and shapes.
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