Quadratic Equations Revision Notes for Grade 12 NSC Matric Mathematics

Quadratic Equations

What is a quadratic equation?

A quadratic equation is a mathematical equation where the highest power of the variable (usually x) is 2. This makes it different from linear equations where the highest power is 1.

The standard form of a quadratic equation is written as:

$ax^2 + bx + c = 0$

where:

a, b, and c are constants (numbers)
a ≠ 0 (if a = 0, the equation becomes linear, not quadratic)
x is the variable we need to solve for

chatImportant

The coefficient 'a' must never equal zero in a quadratic equation. If a = 0, the equation becomes linear (first degree) rather than quadratic (second degree).

Examples of quadratic equations

Here are some examples to help you recognise quadratic equations:

$x^2 + 5x + 6 = 0$
$3x^2 - 7x = 12$
$(3x - 9)(y + 2) = 5x$ (this becomes quadratic after expansion)

infoNote

Some equations may not look quadratic at first, but become quadratic when you rearrange or expand them. Always look for the highest power of the variable after simplification.

Solving quadratic equations

There are three main methods for solving quadratic equations. Each method has its advantages and is useful in different situations.

Method 1: Solving by factorising

Factorising is often the quickest method when the quadratic expression can be broken down into simpler factors.

Steps for solving by factorising:

Step 1: Write the equation in standard form: $ax^2 + bx + c = 0$
Step 2: Factorise the quadratic expression into two brackets
Step 3: Use the zero-product property: If $(A)(B) = 0$ , then either $A = 0$ or $B = 0$
Step 4: Solve each factor equation to find the values of x

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Worked Example: Solving by Factorising

Solve $x^2 - 7x + 10 = 0$

Solution:

The equation is already in standard form
Factorise: $(x - 5)(x - 2) = 0$
Apply zero-product property: either $x - 5 = 0$ or $x - 2 = 0$
Solve: $x = 5$ or $x = 2$

Method 2: Completing the square

Completing the square is useful when you want to rewrite the quadratic in vertex form or when factorising is difficult.

Steps for completing the square:

Step 1: Write the equation in the form: $y = ax^2 + bx + c$
Step 2: Factor out 'a' if $a ≠ 1$
Step 3: Take half of the coefficient of x, square it, then add and subtract this value
Step 4: Rewrite as a perfect square

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Worked Example: Completing the Square

Solve $y = 3x^2 + 12x + 9$

Solution:

Factor out 3: $y = 3(x^2 + 4x) + 9$
Take half of 4, square it: $(4/2)^2 = 4$
Add and subtract 4 inside the bracket: $y = 3(x^2 + 4x + 4 - 4) + 9$
Factorise the perfect square: $y = 3(x + 2)^2 - 3(4) + 9$
Simplify: $y = 3(x + 2)^2 - 12 + 9 = 3(x + 2)^2 - 3$

Method 3: The quadratic formula

When a quadratic equation cannot be factorised easily, use the quadratic formula. This method always works for any quadratic equation.

The quadratic formula:

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

where a, b, and c are the coefficients from the standard form $ax^2 + bx + c = 0$ .

chatImportant

The quadratic formula is your most reliable method - it works for every quadratic equation, even when factorising is impossible or completing the square is too complex.

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Worked Example: Using the Quadratic Formula

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

Solution:

Identify: $a = 1$ , $b = -5$ , $c = 3$
Substitute into formula: $x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(3)}}{2(1)}$
Simplify: $x = \frac{5 \pm \sqrt{25 - 12}}{2} = \frac{5 \pm \sqrt{13}}{2}$
Solutions: $x = \frac{5 + \sqrt{13}}{2}$ or $x = \frac{5 - \sqrt{13}}{2}$

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Worked Example: Rearranging First

Solve $4x^2 - 8x = 7$

Solution:

Write in standard form: $4x^2 - 8x - 7 = 0$
Identify: $a = 4$ , $b = -8$ , $c = -7$
Substitute: $x = \frac{8 \pm \sqrt{64 + 112}}{8} = \frac{8 \pm \sqrt{176}}{8}$
Solutions: $x = \frac{8 + \sqrt{176}}{8}$ or $x = \frac{8 - \sqrt{176}}{8}$
Approximate: $x ≈ 2.66$ or $x ≈ -0.66$

Finding unknown parameters

Sometimes you'll be given that a specific value is a root (solution) of a quadratic equation, and you need to find an unknown parameter.

Method: Substitute the known root into the equation and solve for the unknown parameter.

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Worked Example: Finding Unknown Parameters

Given that $\frac{2}{3}$ is a root of $12x^2 - kx - 8 = 0$ , find k.

Solution:

Substitute $x = \frac{2}{3}$ : $12\left(\frac{2}{3}\right)^2 - k\left(\frac{2}{3}\right) - 8 = 0$
Simplify: $12 \times \frac{4}{9} - \frac{2k}{3} - 8 = 0$
Calculate: $\frac{48}{9} - \frac{2k}{3} - 8 = 0$
Convert to common denominator: $\frac{16}{3} - \frac{2k}{3} - 8 = 0$
Multiply by 3: $16 - 2k - 24 = 0$
Solve: $-2k = 8$ , so $k = -4$

When to use each method

Understanding when to use each method will make you more efficient at solving quadratic equations:

infoNote

Method Selection Guidelines:

Factorising: Use when the quadratic can be easily factored. This is usually the quickest method.
Completing the square: Use when you need the vertex form or when factorising is difficult but you want an exact answer.
Quadratic formula: Use when factorising is not possible or when you need decimal approximations.

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Key Points to Remember:

A quadratic equation has the highest power of x equal to 2 and follows the standard form $ax^2 + bx + c = 0$
Factorising is the fastest method but only works when the expression can be factored easily
Completing the square helps rewrite equations in vertex form and gives exact solutions
The quadratic formula always works, even when other methods fail
Always check your solutions by substituting back into the original equation

Quadratic Equations (Grade 12 NSC Matric Mathematics): Revision Notes