Quadratic Formula Revision Notes for Grade 11 NSC Matric Mathematics

Quadratic Formula

What is the quadratic formula?

The quadratic formula is a powerful mathematical tool used to solve any quadratic equation when other methods like factorisation don't work or are too time-consuming. While factorisation and completing the square are useful techniques, they aren't always practical for every quadratic equation. The quadratic formula provides a reliable method that works for all quadratic equations.

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The quadratic formula was derived by completing the square on the general quadratic equation. While you don't need to memorize the derivation, understanding that it comes from completing the square helps explain why it always works.

When to use the quadratic formula

You should use the quadratic formula when:

The quadratic equation cannot be factorised easily
Completing the square becomes too complex
You need a quick and reliable method to find exact solutions
The quadratic equation has non-integer coefficients

chatImportant

Always check if a quadratic can be factorised first, as factorisation is often quicker when possible. However, the quadratic formula is your reliable backup method that works in every situation.

The quadratic formula

For any quadratic equation in standard form $ax^2 + bx + c = 0$ where $a ≠ 0$ , the solutions are given by:

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

Where:

$a$ is the coefficient of $x^2$
$b$ is the coefficient of $x$
$c$ is the constant term
The symbol $\pm$ means plus or minus, giving us two possible solutions

The discriminant

The expression under the square root, $b^2 - 4ac$ , is called the discriminant. The discriminant tells us about the nature of the roots:

If $b^2 - 4ac > 0$ : Two distinct real roots exist
If $b^2 - 4ac = 0$ : One repeated real root exists
If $b^2 - 4ac < 0$ : No real roots exist (the roots are imaginary)

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This is crucial information because it tells us whether our quadratic equation has real solutions before we even calculate them. Always check the discriminant first - it can save you time!

Step-by-step method

Step 1: Check if the equation can be factorised easily
If factorisation is straightforward, use that method first as it's often quicker.

Step 2: Write the equation in standard form
Rearrange the equation so it equals zero: $ax^2 + bx + c = 0$

Step 3: Identify the coefficients
Carefully identify the values of $a$ , $b$ , and $c$ from your equation.

Step 4: Apply the quadratic formula
Substitute the values into $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Step 5: Simplify and write the final answer
Calculate the discriminant first, then complete the calculation to find both roots.

Worked example 1: Real roots

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Worked Example: Solving with Real Roots

Question: Solve $2x^2 + 3x = 7$ using the quadratic formula.

Solution:

Step 1: Check for factorisation
This equation cannot be factorised easily, so we use the quadratic formula.

Step 2: Write in standard form
$2x^2 + 3x - 7 = 0$

Step 3: Identify coefficients
$a = 2$ , $b = 3$ , $c = -7$

Step 4: Apply the formula
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

$x = \frac{-3 \pm \sqrt{(3)^2 - 4(2)(-7)}}{2(2)}$

$x = \frac{-3 \pm \sqrt{9 + 56}}{4}$

$x = \frac{-3 \pm \sqrt{65}}{4}$

Step 5: Final answer
The two roots are: $x = \frac{-3 + \sqrt{65}}{4}$ or $x = \frac{-3 - \sqrt{65}}{4}$

Worked example 2: No real roots

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Worked Example: When No Real Roots Exist

Question: Find the roots of $f(x) = x^2 - 5x + 8$ .

Solution:

Step 1: Set the function equal to zero
$x^2 - 5x + 8 = 0$

Step 2: Check for factorisation
This cannot be factorised, so we use the quadratic formula.

Step 3: Identify coefficients
$a = 1$ , $b = -5$ , $c = 8$

Step 4: Apply the formula
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

$x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(8)}}{2(1)}$

$x = \frac{5 \pm \sqrt{25 - 32}}{2}$

$x = \frac{5 \pm \sqrt{-7}}{2}$

Step 5: Final answer
Since the discriminant is negative ( $-7 < 0$ ), there are no real roots. The graph of this quadratic function lies entirely above the x-axis.

Worked example 3: Perfect square

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Worked Example: Repeated Root (Perfect Square)

Question: Solve $x^2 - 6x + 9 = 0$ using the quadratic formula.

Solution:

Step 1: Check the equation
$x^2 - 6x + 9 = 0$

Step 2: Identify coefficients
$a = 1$ , $b = -6$ , $c = 9$

Step 3: Apply the formula
$x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(9)}}{2(1)}$

$x = \frac{6 \pm \sqrt{36 - 36}}{2}$

$x = \frac{6 \pm \sqrt{0}}{2}$

$x = \frac{6 \pm 0}{2} = 3$

Step 4: Final answer
There is one repeated root: $x = 3$ . This occurs when the discriminant equals zero.

Common exam tips

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Exam Success Tips

Always write the formula down first before substituting values
Be careful with negative signs, especially when $b$ or $c$ are negative
Check your discriminant first - it saves time if there are no real roots
Simplify your final answers completely, leaving surds in exact form
Remember that you should get two solutions (or one repeated solution)
If asked for decimal approximations, use your calculator only at the final step

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Key Points to Remember:

The quadratic formula works for any quadratic equation: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
The discriminant ( $b^2 - 4ac$ ) determines the nature of roots: positive gives real roots, zero gives one repeated root, negative gives no real roots
Always rearrange to standard form first: $ax^2 + bx + c = 0$
Identify coefficients carefully: watch out for negative signs and missing terms
The formula gives two solutions: use both the plus and minus versions of $\pm$

Quadratic Formula (Grade 11 NSC Matric Mathematics): Revision Notes