JEE Advanced Quadratic Equations Important Questions

From 2018, the JEE exam will be conducted in the online mode. The concept of Quadratic equations forms the basis of algebra in the higher class. This chapter has vast applications and can be combined with a lot of other chapters while being questioned in the actual examination. You can expect around 1 to 2 questions from this chapter in the JEE Advanced paper. 

The answers given to the Quadratic Equations Important Questions JEE Advanced here are filed with numerous short cut techniques and quick tips. This is very much necessary as the question in the JEE Advanced paper expects to only choose the right answer from the given set of answers. Therefore it is important to have speed and accuracy while solving these sums. 

Here is a list of a few important concepts in this chapter based on which questions have been asked repeatedly in the previous year's papers.

• The general form of quadratic equation

• Conditions for nature of roots using discriminant

• Conditions for minimum and maximum value of parabola

• Conditions for minimum and maximum value of a parabola

• Roots of quadratic equation

• The relation between roots and coefficient of the quadratic equation

• The discriminant of quadratic equations.

The quadratic equation is a second-degree equation. It is generally expressed in the form of ax²+bx+c = 0, where a, b, c are real numbers and a ≠ 0. For example, x²+2x+1 = 0. An algebraic expression with multiple terms is called a polynomial. When the quadratic polynomial is equated to zero, it is called a quadratic polynomial.

Roots of a Quadratic Equation:

If a value of x satisfies the quadratic equation, then they are called roots of the quadratic equation. They are also called zeros of solutions of the equation. An equation can be solved either by the factorisation method where you factorize the equation and equate it to zero or by using the quadratic formula. They both give solutions to the equation.

Sum and Product of Roots:

If α and β are said to be the roots of the quadratic equation ax2+bx+c = 0,

Then the product of the roots is αβ = c/a

The sum of the roots is α + β = -b/a

Nature of Roots:

 D in quadratic equation denotes discriminant

• If D is equal to 0, the roots of the equation are real and equal.

• If D is greater than 0, the roots of the equation are real and unequal.

• If D is less than 0, the roots of the equation are imaginary and unequal.

• If D is greater than 0 and D is a perfect square, the roots of the equation are rational and unequal.

• If D is greater than 0 and D is not a perfect square, the roots of the equation are irrational and unequal.

Graph of a Quadratic Equation:

Consider a quadratic equation ax²+bx+c = 0, where a, b, and c are real and

 a is not equal to 0. The figure we see on the graph of a quadratic equation is a parabola.

• Case 1: If a>0, D > 0: the graph of the equation is concave upwards and intersects the x-axis at two points. Then the equation will have two real roots. The curve is always above the x-axis

• Case 2: If a>0, D = 0:  the graph of the equation curves upwards and intersects the x-axis at one point. The equation will have two equal roots

• Case 3: If a>0, D < 0: the graph has a curve upwards in a concave shape.  The curve does not intersect the x-axis and the equation has imaginary roots.

• Case 4: If a<0, D>0:  the graph has a curve downwards in a concave shape and intersects the x-axis at two points. The roots of the equation are two real roots.

• Case 5: If a<0, D=0: the parabola is curved downwards and intersects the x-axis at one point. The equation will have two equal roots.

• Case 6: If a<0, D<0: the parabola is concave downwards and does not intersect the x-axis at any point. The equation will have imaginary roots.

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Most Important Formulas of Quadratic Equations for JEE Advanced

1. Quadratic Formula:

• $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

2. Nature of Roots:

• If $b^2 - 4ac > 0$, the roots are real and distinct.

• If $b^2 - 4ac = 0$, the roots are real and equal.

• If $b^2 - 4ac < 0$, the roots are complex conjugates.

3. Sum and Product of Roots:

• Sum of roots, $\alpha + \beta = -\frac{b}{a}$

• Product of roots, $\alpha \cdot \beta = \frac{c}{a}$

4. Quadratic Equation in Vertex Form:

• $y = a(x - h)^2 + k$, where (h,k) is the vertex of the parabola.

5. Nature of Quadratic Expression:

• If a > 0, the parabola opens upwards, and the vertex is the minimum point.

• If a < 0, the parabola opens downwards, and the vertex is the maximum point.

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