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CAT 2025 Lesson : Quadratic Equations - Discriminant & Graphical Representation

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3.1 Discriminant & Nature of Roots

The roots of an equation are the solution or the values of x where the quadratic equation holds good. Every quadratic equation has two roots.

For a general quadratic equation written as
ax2+bx+c=0ax^{2} + bx + c = 0,

x=b+b24ac2a,bb24ac2ax = \dfrac{-b + \sqrt{b^{2} - 4ac}}{2a}, \dfrac{-b - \sqrt{b^{2} - 4ac}}{2a}

Note that the two roots are generally read together as
x=b±b24ac2a \bm{x = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a}} .

The expression inside the square root is commonly called the discriminant, i.e.
D=b24ac\bm{D = b^{2} - 4ac}.

∴ The roots can be written as
x=b±D2ax = \dfrac{-b \pm \sqrt{D}}{2a}

As the discriminant is inside a square root, the nature of roots can be defined as follows.

Discriminant Nature of Roots
D>0D \gt 0 Two real and distinct roots.
D=0D = 0 Two real and equal roots
D<0D \lt 0 Two imaginary roots


Example 2

Which of the following equations have real and distinct roots?

(1)
x22x+7=0x^{2} - 2x + 7 = 0
(2)
x22x7=0x^{2} - 2x - 7 = 0
(3)
4x212x+9=04x^{2} - 12x + 9 = 0
(4)
2x2+5x+2=02x^{2} + 5x + 2 = 0

Solution

The following discriminants help us identify the nature of roots for each of the equations.

Equation Discriminant Nature of Roots
11 (2)24×1×7=24(-2)^{2} - 4 \times 1 \times 7 = -24 Imaginary roots
22 (2)24×1×7=32(-2)^{2} - 4 \times 1 \times -7 = 32 Real and Distinct roots
33 (12)24×4×9=0(-12)^{2} - 4 \times 4 \times 9 = 0 Real and Equal roots
44 (5)24×2×2=9(5)^{2} - 4 \times 2 \times 2 = 9 Real and Distinct roots


Answer: (2)(2) and (4)(4)


3.2 Graphical Representation

The following graphical representation of quadratic expressions will help you better understand the roots of the equations. The curves are for the expression
f(x)=ax2+bx+cf(x) = ax^{2} + bx + c.

When the curve cuts the
xx-axis, then at these points f(x)=0f(x) = 0. The xx-values at these points are the roots of the quadratic expression.

11) All quadratic expressions are either U-shaped (when a > 0) or inverted U-shaped (when a < 0)
22) In a U-shaped curve, we can find the minimum value that the quadratic expression can attain. And, in an inverted U-shaped curve, we can find the maximum value that the quadratic expression can attain. This is covered later in this lesson.
33) When D >\bm{\gt} 0, the 22 distinct real roots are points at which the curve cuts the xx-axis.
44) When D = 0, the real and equal root is the point at which the curve touches the x-axis.
55) When D <\bm{\lt} 0, the curve does not cut the x-axis and, therefore, the roots are imaginary.

f(x)=ax2+bx+c\bm{f(x) = ax^{2} + bx + c} a>0\bm{a \gt 0} a<0\bm{a \lt 0}
D >\gt 0
(Real & Distinct roots)
D == 0
(Real & Equal roots)
D <\lt 0
(Imaginary roots)


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