Nature of Roots Revision Notes for Grade 11 NSC Matric Mathematics

Nature of Roots

What is the discriminant?

The discriminant is a mathematical tool that helps us determine what types of roots a quadratic equation will have without actually solving the equation.

Definition: The discriminant is the expression under the square root sign in the quadratic formula.

Formula: For a quadratic equation in standard form $ax^2 + bx + c = 0$ , the discriminant is:

$\Delta = b^2 - 4ac$

Where $\Delta$ (delta) is the Greek symbol used to represent the discriminant.

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The discriminant appears in the quadratic formula and is the key to understanding root behavior. By calculating this single value, we can predict whether our quadratic equation will have real or imaginary solutions, and if real, whether they are equal or unequal, rational or irrational.

The discriminant appears in the quadratic formula: $x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

How the discriminant determines root types

The value of the discriminant tells us important information about the nature of the roots. The word 'nature' refers to the types of numbers the roots can be - namely real, rational, irrational or imaginary.

The discriminant acts like a decision-making tool that classifies roots into different categories based on its value.

Classification system for roots

Based on the discriminant value, we can classify roots into the following categories:

chatImportant

Case 1: $\Delta < 0$ (discriminant is negative)

Roots are imaginary (non-real)
The expression under the square root is negative
These roots are beyond the scope of this course level

chatImportant

Case 2: $\Delta \geq 0$ (discriminant is non-negative)

Roots are real
This case splits into two sub-cases:

Sub-case 2a: $\Delta = 0$ (discriminant equals zero)

Roots are real and equal
There is only one root (or we say there are two identical roots)
The parabola touches the x-axis at exactly one point

Sub-case 2b: $\Delta > 0$ (discriminant is positive)

Roots are real and unequal
There are two different roots

When we have real and unequal roots ( $\Delta > 0$ ), we can further classify them:

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If $\Delta$ is a perfect square of a rational number:

Roots are rational
Can be expressed as exact fractions or integers

If $\Delta$ is not a perfect square of a rational number:

Roots are irrational
Must be expressed in decimal or surd form

Step-by-step method to determine nature of roots

Follow these steps to determine the nature of roots for any quadratic equation:

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Step-by-Step Process:

Step 1: Check that the equation is in standard form $ax^2 + bx + c = 0$

Step 2: Identify the coefficients $a$ , $b$ , and $c$

Step 3: Calculate the discriminant using $\Delta = b^2 - 4ac$

Step 4: Apply the classification rules:

If $\Delta < 0$ : roots are imaginary
If $\Delta = 0$ : roots are real and equal
If $\Delta > 0$ $Δ > 0$ : roots are real and unequal
- Check if $\Delta$ is a perfect square to determine if roots are rational or irrational

Step 5: State your conclusion clearly

Worked examples

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Worked Example 1: Determining if roots are irrational

Question: Show that the roots of $x^2 - 2x - 7 = 0$ are irrational.

Solution:

Step 1: The equation is already in standard form $ax^2 + bx + c = 0$

Step 2: Identify coefficients

$a = 1$
$b = -2$
$c = -7$

Step 3: Calculate the discriminant

$\Delta = b^2 - 4ac$
$= (-2)^2 - 4(1)(-7)$
$= 4 + 28$
$= 32$

Step 4: Interpret the result

Since $32 > 0$ and 32 is not a perfect square, we can conclude the roots are real, unequal and irrational.

Step 5: Final answer The roots are real, unequal and irrational.

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Worked Example 2: Finding values for equal roots

Question: For which value(s) of $k$ will the roots of $6x^2 + 6 = 4kx$ be real and equal?

Solution:

Step 1: Rearrange to standard form

$6x^2 - 4kx + 6 = 0$

Step 2: Identify coefficients

$a = 6$
$b = -4k$
$c = 6$

Step 3: For real and equal roots, $\Delta = 0$

$\Delta = b^2 - 4ac = 0$
$(-4k)^2 - 4(6)(6) = 0$
$16k^2 - 144 = 0$

Step 4: Solve for $k$

$16k^2 = 144$
$k^2 = 9$
$k = 3 \text{ or } k = -3$

Step 5: Verify the answers

For $k = 3$ : $6x^2 - 12x + 6 = 0$ gives $(x-1)^2 = 0$ , so $x = 1$ (equal roots) ✓
For $k = -3$ : $6x^2 + 12x + 6 = 0$ gives $(x+1)^2 = 0$ , so $x = -1$ (equal roots) ✓

Final answer: $k = 3$ or $k = -3$

lightbulbExample

Worked Example 3: Proving roots are always real

Question: Show that the roots of $(x + h)(x + k) = 4d^2$ are real for all real values of $h$ , $k$ and $d$ .

Solution:

Step 1: Expand to standard form

$(x + h)(x + k) = 4d^2$
$x^2 + hx + kx + hk = 4d^2$
$x^2 + (h + k)x + (hk - 4d^2) = 0$

Step 2: Identify coefficients

$a = 1$
$b = h + k$
$c = hk - 4d^2$

Step 3: Calculate the discriminant

$\Delta = b^2 - 4ac$
$= (h + k)^2 - 4(1)(hk - 4d^2)$
$= h^2 + 2hk + k^2 - 4hk + 16d^2$
$= h^2 - 2hk + k^2 + 16d^2$
$= (h - k)^2 + (4d)^2$

Step 4: Interpret the result

Since $(h - k)^2 \geq 0$ and $(4d)^2 \geq 0$ for all real values, we have:
$(h - k)^2 + (4d)^2 \geq 0$
Therefore $\Delta \geq 0$

Step 5: Final answer

Since $\Delta \geq 0$ , the roots are real for all real values of $h$ , $k$ and $d$ .

bookmarkSummary

Key Points to Remember:

The discriminant formula is $\Delta = b^2 - 4ac$ - this is your starting point for determining root types
Negative discriminant means imaginary roots, zero discriminant means equal roots, positive discriminant means unequal real roots
For real unequal roots, check if the discriminant is a perfect square to determine if roots are rational or irrational
Always write your equation in standard form first ( $ax^2 + bx + c = 0$ ) before identifying coefficients
When asked to find parameter values for specific root types, set the discriminant equal to the appropriate value and solve

Nature of Roots (Grade 11 NSC Matric Mathematics): Revision Notes

Nature of Roots

What is the discriminant?

How the discriminant determines root types

Classification system for roots

Step-by-step method to determine nature of roots

Worked examples

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