Solving Quadratic Equations Revision Notes for Grade 10 NSC Matric Mathematics

Solving Quadratic Equations

What is a quadratic equation?

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A quadratic equation is an equation where the highest power (exponent) of the variable is 2. This means the variable appears as a square term like x².

The standard form of a quadratic equation is: $ax^2 + bx + c = 0$

where a, b, and c are constants, and a ≠ 0.

Examples of quadratic equations include:

$2x^2 + 2x = 1$
$3x^2 + 2x - 1 = 0$
$0 = -2x^2 + 4x - 2$

How quadratic equations differ from linear equations

Understanding the key differences between linear and quadratic equations is essential for your exam preparation:

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Key Differences:

Linear equations have only one solution
Quadratic equations can have at most two solutions
In special cases, a quadratic equation may have only one solution or no real solutions

This difference occurs because when you factorise a quadratic equation, you typically get two factors that can each equal zero.

Method for solving quadratic equations by factorisation

The most common method for solving quadratic equations at this level is factorisation.

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Five-Step Method for Solving Quadratic Equations:

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

Step 2: Divide the entire equation by any common factor of the coefficients to simplify (if needed)

Step 3: Factorise the quadratic expression to get the form $(rx + s)(ux + v) = 0$

Step 4: Use the zero product property: if $a \times b = 0$ , then either $a = 0$ or $b = 0$

Therefore: $(rx + s) = 0$ or $(ux + v) = 0$

This gives you: $x = -\frac{s}{r}$ or $x = -\frac{v}{u}$

Step 5: Check your answers by substituting back into the original equation

Worked example 1: Standard quadratic equation

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Worked Example: Solving a Standard Quadratic Equation

Question: Solve $3x^2 + 2x - 1 = 0$

Solution:

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

Step 2: Factorise the expression

$3x^2 + 2x - 1 = (x + 1)(3x - 1) = 0$

Step 3: Solve each factor

$(x + 1) = 0$ OR $(3x - 1) = 0$

$x = -1$ OR $x = \frac{1}{3}$

Step 4: Check both solutions by substituting back into the original equation

Step 5: Write the final answer

The solutions are $x = -1$ or $x = \frac{1}{3}$

Worked example 2: Perfect square quadratic

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Worked Example: Perfect Square Quadratic

Question: Find the roots of $0 = -2x^2 + 4x - 2$

Solution:

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

Step 2: Divide by the common factor -2

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

$x^2 - 2x + 1 = 0$

Step 3: Factorise

$(x - 1)(x - 1) = 0$

$(x - 1)^2 = 0$

Step 4: This is a perfect square - both factors are identical

This means there is only one solution: $x - 1 = 0$

Therefore: $x = 1$

Step 5: Check and write the final answer

The solution is $x = 1$

Special cases in quadratic equations

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Sometimes quadratic equations have special characteristics:

Perfect square quadratics like $(x - 1)^2 = 0$ have only one solution because both factors are the same
Some quadratic equations may have no real solutions when factorisation is not possible with real numbers
The number of solutions depends on the nature of the quadratic expression

Exam tips

Here are essential strategies for tackling quadratic equations in your exams:

Always check your solutions by substituting back into the original equation
Remember that quadratic equations typically have two solutions
If you get the same solution twice, you have a perfect square situation
Write your final answers clearly, showing both solutions when they exist
Practice identifying when an equation is already in standard form versus when you need to rearrange it first

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Key Points to Remember:

A quadratic equation has the variable raised to the power of 2 as its highest term
Quadratic equations can have at most two solutions, unlike linear equations which have exactly one
Use factorisation by finding two expressions that multiply to give the quadratic
Apply the zero product property: if two factors multiply to zero, at least one factor must be zero
Always check your answers by substituting back into the original equation

Solving Quadratic Equations (Grade 10 NSC Matric Mathematics): Revision Notes