Nature Of Roots & The Quadratic Formula

Student can determine the nature of roots using the discriminant and calculate roots using the quadratic formula.

8 lessons

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The Quadratic Formula

Introduce the quadratic formula for finding roots directly.

Historically, mathematicians like Sridharacharya derived a powerful formula for solving quadratic equations by a method known as completing the square. Today, this is simply known as the Quadratic Formula.

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Formula Card: Quadratic Formula

Key formula and variables.

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

Variables

$a, b, c$ are the coefficients of $a x^{2} + b x + c = 0$ .
$a \neq = 0$ .

Real Roots

$b^{2} - 4 a c \geq 0$ → roots are real.
$b^{2} - 4 a c < 0$ → no real roots.

Example

For

2 x^{2} - 5 x + 3 = 0

x = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 ( 2 ) ( 3 )}{2 ( 2 )}

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The Discriminant

Define b^2 - 4ac as the discriminant determining root nature.

Look closely at the expression sitting inside the square root of the quadratic formula: $b^2 - 4ac$ .

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Nature of Roots Summary

Compare the 3 cases of the discriminant.

$b^{2} - 4 a c > 0$

Two distinct real roots
Roots take the full form: $\frac{- b \pm b ^{2} - 4 a c}{2 a}$

$b^{2} - 4 a c = 0$

Two equal real roots (coincident)
Since the root vanishes, both roots equal: $- \frac{b}{2 a}$

$b^{2} - 4 a c < 0$

No real roots
Cannot calculate the square root of a negative value in real numbers.

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Example 7: Finding the Discriminant

Solve Example 7 to find root nature.

Problem. Find the discriminant of the quadratic equation $2x^2 - 4x + 3 = 0$ , and hence find the nature of its roots.

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Example 8: Using Discriminant for Feasibility

Check if a physical situation is possible.

Problem. A pole has to be erected on the boundary of a circular park of diameter 13m in such a way that the difference of its distances from two diametrically opposite gates A and B is 7m. Is it possible to do so? If yes, at what distances from the two gates?

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Determine Nature of Roots

Practice calculating the discriminant.

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What is the nature of the roots for the quadratic equation $2x^2 - 3x + 5 = 0$ ?

← previousSolving By Factorisation next →Chapter Review: Quadratic Equations

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The Discriminant

Define b^2 - 4ac as the discriminant determining root nature.

Look closely at the expression sitting inside the square root of the quadratic formula: $b^2 - 4ac$ .

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Example 7: Finding the Discriminant

Solve Example 7 to find root nature.

Problem. Find the discriminant of the quadratic equation $2x^2 - 4x + 3 = 0$ , and hence find the nature of its roots.

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quiz

Determine Nature of Roots

Practice calculating the discriminant.

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What is the nature of the roots for the quadratic equation $2x^2 - 3x + 5 = 0$ ?

Nature Of Roots & The Quadratic Formula

The Quadratic Formula

Formula Card: Quadratic Formula

The Discriminant

Nature of Roots Summary

b2−4ac>0

b2−4ac=0

b2−4ac<0

Example 7: Finding the Discriminant

Example 8: Using Discriminant for Feasibility

Determine Nature of Roots

The Quadratic Formula

Formula Card: Quadratic Formula

The Discriminant

Nature of Roots Summary

b2−4ac>0

b2−4ac=0

b2−4ac<0

Example 7: Finding the Discriminant

Example 8: Using Discriminant for Feasibility

Determine Nature of Roots

$b^{2} - 4 a c > 0$

$b^{2} - 4 a c = 0$

$b^{2} - 4 a c < 0$

$b^{2} - 4 a c > 0$

$b^{2} - 4 a c = 0$

$b^{2} - 4 a c < 0$