Maths934 views·Updated 27 Jul 2026·6 pages

Mastering Quadratic Equations

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Quadratic equations might seem scary at first, but they're actually...

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Factorising Simple Quadratic Equations

The easiest way to solve quadratic equations is by factorising when the coefficient of x² is 1. For an equation like x² + 5x + 6 = 0, you need to find two numbers that multiply to give c (the constant term) and add to give b (the coefficient of x).

Take x² + 5x + 6 = 0 as an example. List all the factor pairs of 6: (1,6), (2,3), $-1,-6$ , $-2,-3$ . The pair that adds up to 5 is 2 and 3, so you get $x + 2$$x + 3$ = 0.

Once you've factorised, set each bracket equal to zero: $x + 2$ = 0 gives x = -2, and $x + 3$ = 0 gives x = -3. Always check your answers by substituting back into the original equation!

Top Tip: When factorising, remember that if c is positive and b is positive, both numbers in your factor pair will be positive. If c is positive and b is negative, both numbers will be negative.

of 6

Factorising When the Coefficient of x² Isn't 1

When you have something like 2x² + 7x + 6 = 0, factorising gets a bit trickier but follows the same principle. You need to find two numbers that multiply to give ac (a times c) and add to give b.

For 2x² + 7x + 6 = 0, multiply a and c: 2 × 6 = 12. Find factor pairs of 12 that add to 7. The pair 3 and 4 works perfectly.

Split the middle term using these numbers, then factorise by grouping. You'll end up with $x + 2$$2x + 3$ = 0, giving you x = -2 or x = -3/2. This method takes practice, but once you get the hang of it, it becomes second nature.

Quick Check: Always multiply out your brackets to verify you get back to the original equation - it's a great way to catch silly mistakes!

of 6

More Complex Factorising Examples

These trickier quadratic equations follow the same pattern but with bigger numbers. For 6x² + 11x + 3 = 0, multiply a and c: 6 × 3 = 18. You need factor pairs of 18 that add to 11.

The factors of 18 include (2,9) and (3,6). Since 2 + 9 = 11, you split 11x into 2x + 9x. This gives you 6x² + 2x + 9x + 3, which factors as $2x + 3$$3x + 1$ = 0.

Solving gives x = -3/2 or x = -1/3. The same method works for any quadratic - find the right factor pair, split the middle term, then factorise by grouping. Practice with different examples helps you spot the patterns more quickly.

Memory Aid: Write down all factor pairs systematically - don't try to work them out in your head, as you might miss the right combination!

of 6

Building Confidence with Factorising

Let's tackle 4x² + 19x + 12 = 0 using the same reliable method. First, calculate ac: 4 × 12 = 48. Now list the factor pairs of 48 systematically.

The factors include (1,48), (2,24), (3,16), (4,12), and (6,8). Since we need them to add to 19, the pair (3,16) works perfectly because 3 + 16 = 19.

Split 19x into 3x + 16x, giving 4x² + 3x + 16x + 12. Factor by grouping to get $x + 4$$4x + 3$ = 0. This gives solutions x = -4 or x = -3/4. See how the same method works every time?

Confidence Booster: Once you've mastered this systematic approach, you can solve any factorable quadratic equation - no matter how complex it looks!

of 6

Completing the Square Method

Completing the square is brilliant when factorising doesn't work easily. For x² + 10x + 10, take half the coefficient of x (which gives 5), then write $x + 5$ ² - 25 + 10 = $x + 5$ ² - 15.

To solve $x + 5$ ² - 15 = 0, rearrange to get $x + 5$ ² = 15, then take the square root: x + 5 = ±√15. Therefore x = -5 ± √15.

When the coefficient of x² isn't 1, like in 2x² - 6x - 3, factor out the 2 first: 2 $x² - 3x$ - 3. Complete the square inside the brackets: 2 $x - 3/2$ ² - 2 $9/4$ - 3 = 2 $x - 3/2$ ² - 15/2.

Why It's Useful: Completing the square also tells you the minimum or maximum point of the quadratic graph - super handy for coordinate geometry questions!

of 6

The Quadratic Formula - Your Ultimate Backup

The quadratic formula x = $-b ± √(b² - 4ac)$ /2a works for absolutely any quadratic equation. When factorising gets messy or completing the square is awkward, this formula saves the day.

For any equation in the form ax² + bx + c = 0, just substitute your values. Take 2x² + 6x + 3 = 0: a = 2, b = 6, c = 3. Plugging into the formula gives x = $-6 ± √(36 - 24)$ /4 = $-6 ± √12$ /4.

This simplifies to x = $-6 ± 2√3$ /4 = $-3 ± √3$ /2. You can use your calculator to get decimal answers: approximately -0.63 and -2.37. The quadratic formula never lets you down!

Pro Tip: Learn to recognise when b² - 4ac (the discriminant) is negative - this means there are no real solutions, which is important information for your answer!

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Maths934 views·Updated 27 Jul 2026·6 pages

Mastering Quadratic Equations

issy@issy.studies

Add to Google sources

Quadratic equations might seem scary at first, but they're actually quite straightforward once you know the tricks. There are several brilliant methods to solve them, from simple factorising to completing the square and using the quadratic formula.

of 6

Factorising Simple Quadratic Equations

Take x² + 5x + 6 = 0 as an example. List all the factor pairs of 6: (1,6), (2,3), $-1,-6$ , $-2,-3$ . The pair that adds up to 5 is 2 and 3, so you get $x + 2$$x + 3$ = 0.

Once you've factorised, set each bracket equal to zero: $x + 2$ = 0 gives x = -2, and $x + 3$ = 0 gives x = -3. Always check your answers by substituting back into the original equation!

Top Tip: When factorising, remember that if c is positive and b is positive, both numbers in your factor pair will be positive. If c is positive and b is negative, both numbers will be negative.

of 6

Factorising When the Coefficient of x² Isn't 1

For 2x² + 7x + 6 = 0, multiply a and c: 2 × 6 = 12. Find factor pairs of 12 that add to 7. The pair 3 and 4 works perfectly.

Quick Check: Always multiply out your brackets to verify you get back to the original equation - it's a great way to catch silly mistakes!

of 6

More Complex Factorising Examples

These trickier quadratic equations follow the same pattern but with bigger numbers. For 6x² + 11x + 3 = 0, multiply a and c: 6 × 3 = 18. You need factor pairs of 18 that add to 11.

The factors of 18 include (2,9) and (3,6). Since 2 + 9 = 11, you split 11x into 2x + 9x. This gives you 6x² + 2x + 9x + 3, which factors as $2x + 3$$3x + 1$ = 0.

Memory Aid: Write down all factor pairs systematically - don't try to work them out in your head, as you might miss the right combination!

of 6

Building Confidence with Factorising

Let's tackle 4x² + 19x + 12 = 0 using the same reliable method. First, calculate ac: 4 × 12 = 48. Now list the factor pairs of 48 systematically.

The factors include (1,48), (2,24), (3,16), (4,12), and (6,8). Since we need them to add to 19, the pair (3,16) works perfectly because 3 + 16 = 19.

Split 19x into 3x + 16x, giving 4x² + 3x + 16x + 12. Factor by grouping to get $x + 4$$4x + 3$ = 0. This gives solutions x = -4 or x = -3/4. See how the same method works every time?

Confidence Booster: Once you've mastered this systematic approach, you can solve any factorable quadratic equation - no matter how complex it looks!

of 6

Completing the Square Method

Completing the square is brilliant when factorising doesn't work easily. For x² + 10x + 10, take half the coefficient of x (which gives 5), then write $x + 5$ ² - 25 + 10 = $x + 5$ ² - 15.

To solve $x + 5$ ² - 15 = 0, rearrange to get $x + 5$ ² = 15, then take the square root: x + 5 = ±√15. Therefore x = -5 ± √15.

When the coefficient of x² isn't 1, like in 2x² - 6x - 3, factor out the 2 first: 2 $x² - 3x$ - 3. Complete the square inside the brackets: 2 $x - 3/2$ ² - 2 $9/4$ - 3 = 2 $x - 3/2$ ² - 15/2.

Why It's Useful: Completing the square also tells you the minimum or maximum point of the quadratic graph - super handy for coordinate geometry questions!

of 6

The Quadratic Formula - Your Ultimate Backup

The quadratic formula x = $-b ± √(b² - 4ac)$ /2a works for absolutely any quadratic equation. When factorising gets messy or completing the square is awkward, this formula saves the day.

This simplifies to x = $-6 ± 2√3$ /4 = $-3 ± √3$ /2. You can use your calculator to get decimal answers: approximately -0.63 and -2.37. The quadratic formula never lets you down!

Pro Tip: Learn to recognise when b² - 4ac (the discriminant) is negative - this means there are no real solutions, which is important information for your answer!

We thought you’d never ask...

You can download the app from Google Play Store and Apple App Store.

That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.

Students love us — and so will you.

4.6/5App Store

4.7/5Google Play

Stefan SiOS user

Samantha KlichAndroid user

AnnaiOS user

Factorising Simple Quadratic Equations

Factorising When the Coefficient of x² Isn't 1

More Complex Factorising Examples

Building Confidence with Factorising

Completing the Square Method

The Quadratic Formula - Your Ultimate Backup

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