Summary Revision Notes for Grade 11 NSC Matric Mathematics

Summary

This summary covers the fundamental concepts and techniques for solving equations and inequalities. Understanding these core principles will help you tackle various types of mathematical problems with confidence.

Zero product law

The zero product law is a fundamental principle used when solving equations. This law states that if the product of two or more factors equals zero, then at least one of the factors must equal zero.

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Definition: If $a \times b = 0$ , then $a = 0$ and/or $b = 0$ .

This principle is particularly useful when solving quadratic equations that can be factorised. When you have an equation in the form $(x - p)(x - q) = 0$ , you can apply the zero product law to find that either $x - p = 0$ or $x - q = 0$ , giving you the solutions $x = p$ or $x = q$ .

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Worked Example: Applying the Zero Product Law

Solve $(x - 3)(x + 5) = 0$

Using the zero product law:

Either $x - 3 = 0$ , so $x = 3$
Or $x + 5 = 0$ , so $x = -5$

Therefore, the solutions are $x = 3$ and $x = -5$ .

Quadratic formula

The quadratic formula provides a method to find the roots of any quadratic equation. This formula works for all quadratic equations, regardless of whether they can be easily factorised.

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Formula: For a quadratic equation in the form $ax^2 + bx + c = 0$ , the solutions are:

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

The quadratic formula is derived by completing the square on the general quadratic equation. It's essential to remember that $a \neq 0$ (otherwise the equation wouldn't be quadratic).

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Worked Example: Using the Quadratic Formula

Solve $2x^2 + 3x - 5 = 0$

Here, $a = 2$ , $b = 3$ , and $c = -5$ .

$x = \frac{-3 \pm \sqrt{3^2 - 4(2)(-5)}}{2(2)}$
$x = \frac{-3 \pm \sqrt{9 + 40}}{4}$
$x = \frac{-3 \pm \sqrt{49}}{4}$
$x = \frac{-3 \pm 7}{4}$

So $x = \frac{-3 + 7}{4} = 1$ or $x = \frac{-3 - 7}{4} = -\frac{5}{2}$

Discriminant

The discriminant is the expression under the square root sign in the quadratic formula. It provides valuable information about the nature of the roots without actually calculating them.

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Formula: $\Delta = b^2 - 4ac$

The discriminant determines whether the roots of a quadratic equation are real or non-real, and if real, whether they are equal or unequal.

Understanding the discriminant values

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When $\Delta < 0$ (negative discriminant):

The roots are non-real (complex numbers)
The parabola does not cross the x-axis
There are no real solutions

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When $\Delta = 0$ (zero discriminant):

The roots are real and equal
The parabola touches the x-axis at exactly one point
There is one repeated real solution

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When $\Delta > 0$ (positive discriminant):

The roots are real and unequal
The parabola crosses the x-axis at two points
There are two distinct real solutions

Rational vs irrational roots

When $\Delta > 0$ , you can further determine whether the roots are rational or irrational:

If $\Delta$ equals a perfect square (squared rational number), the roots are rational
If $\Delta$ is not a perfect square, the roots are irrational

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Worked Example: Analyzing the Discriminant

For $x^2 - 4x + 3 = 0$ :

$\Delta = (-4)^2 - 4(1)(3) = 16 - 12 = 4$

Since $\Delta = 4 > 0$ and $4 = 2^2$ (a perfect square), the equation has two rational, unequal roots.

Problem-solving strategies

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When approaching equations and inequalities problems:

Identify the type of equation - linear, quadratic, or other forms
Choose the appropriate method - factoring, quadratic formula, or completing the square
Check your solutions by substituting back into the original equation
Consider the domain - ensure solutions don't make denominators zero

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Common exam tips:

Always check if a quadratic can be factorised before using the quadratic formula
When dealing with fractions, multiply through by the common denominator first
Remember that $\sqrt{a^2} = |a|$ , not just $a$
Pay attention to domain restrictions in rational equations

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Key Points to Remember:

Zero product law: If $a \times b = 0$ , then at least one factor equals zero
Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ solves any quadratic equation
Discriminant: $\Delta = b^2 - 4ac$ tells you about the nature of roots before solving
Negative discriminant means no real solutions; zero discriminant means one repeated solution; positive discriminant means two real solutions
Always check your solutions in the original equation to verify they are correct

Summary (Grade 11 NSC Matric Mathematics): Revision Notes

Summary

Zero product law

Quadratic formula

Discriminant

Understanding the discriminant values

Rational vs irrational roots

Problem-solving strategies

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