Solving Quadratic Equations by Factorising
Solving quadratic equations by factorising means writing x² + 5x + 6 = 0 as (x + 2)(x + 3) = 0. You find two numbers that multiply to 6 and add to 5. Then set each bracket to zero to get x = -2 and x = -3.
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Preparing the Equation
- Make sure the equation is written as
- Rearrange first if needed so the equation is equal to zero
Factorising the Quadratic
- Find two numbers m and n that multiply to c and add to b
- Write the equation as
- Example:
Finding the Solutions
- Set each bracket equal to zero: or
- Solve to find both values of x: and
Practice Questions
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What are the solutions for the equation ?
Correct! 🎉 +10 pointsNot quite right
The equation can be solved by setting each bracket equal to zero. This gives and , leading to and .
What are the solutions for the equation ?
Correct! 🎉 +10 pointsNot quite right
The equation can be solved by setting each bracket equal to zero. This gives and , leading to and .
Factorise the quadratic equation: .
Correct! 🎉 +20 pointsNot quite right
To factorise , we need to find two numbers that multiply to give 10 and add up to 7. These numbers are 2 and 5, so the factorised form is .
Factorise the quadratic equation: .
Correct! 🎉 +20 pointsNot quite right
To factorise , we need to find two numbers that multiply to give 28 and add up to −11. These numbers are −4 and −7, so the factorised form is .
Factorise the quadratic equation: .
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To factorise , we need to find two numbers that multiply to −18 and add up to −3. These numbers are −6 and 3, so the factorised form is .
Solve this quadratic equation using factorisation: .
Correct! 🎉 +30 pointsNot quite right
First divide the equation by 5 to simplify it to . Then, find two numbers that multiply to −16 and add to 6. These numbers are −2 and 8, so the factorised form is . Therefore, the solutions are and .
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Students Also Ask
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You need two numbers that multiply together to give the constant term. The same two numbers must add together to give the coefficient of x. For x² + 5x + 6, the numbers 2 and 3 work. They multiply to give 6 and add to give 5.
Keep the sign when you list the factor pairs. A negative constant term means one number is positive and the other is negative. For x² + 3x - 18, use the pair -3 and 6. They multiply to give -18 and add to give 3. So it factorises to (x - 3)(x + 6).
The factorised form is equivalent to the original equation, so it has exactly the same solutions. Writing the quadratic as two brackets makes the solutions easy to read off. Each bracket gives one value of x that makes the equation equal to 0.
Factorising works best when the coefficient of x² is 1 and the other numbers are integers. It will not always work, so some quadratic equations need more sophisticated methods. Even so, it is quick when it does work, so it is worth trying first.
Expand your two brackets and check that you get back the original equation. For example, expand (x + 2)(x + 3) to get x² + 3x + 2x + 6. This simplifies to x² + 5x + 6. If it matches, your factorising is correct.