Solving By Factorisation

Student can find the roots of a quadratic equation by factorizing (splitting the middle term).

9 lessons

Preview · progress not saved — Chapter 1 is free

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Concept

Roots and Zeroes

Explain roots of an equation vs zeroes of a polynomial.

A quadratic equation looks like $ax^2 + bx + c = 0$ . When we solve it, we are looking for values of $x$ that make the equation true.

If we replace $x$ with a real number $\alpha$ , and it perfectly balances the equation so that $a\alpha^2 + b\alpha + c = 0$ , we call $\alpha$ a root of the quadratic equation.

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Method

How to Split the Middle Term

Explain the core logic of finding two numbers to split the 'bx' term.

To solve $ax^2 + bx + c = 0$ by factorisation, we split the middle term ( $bx$ ) into two parts.

But we can't just pick any two numbers that add up to $b$ . There is a strict rule: the two numbers must multiply to equal exactly $(a \times c)$ .

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Flowchart

Factorisation Process

A step-by-step flowchart for splitting the middle term.

polished process flowchart, rounded card-style nodes, pastel gradient fills, elegant sans-serif typography, generous whi…

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Example

Example 3: Splitting the Middle Term

Solving 2x^2 - 5x + 3 = 0.

Problem. Find the roots of the equation $2x^2 - 5x + 3 = 0$ , by factorisation.

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Example

Example 5: Equations with Square Roots

Solving 3x^2 - 2\sqrt{6}x + 2 = 0.

Problem. Find the roots of the quadratic equation $3x^2 - 2\sqrt{6}x + 2 = 0$ .

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reading

Your Turn: Factorisation

Faded example for solving x^2 - 3x - 10 = 0.

0 of 3 blanks filled

To solve $x^2 - 3x - 10 = 0$ , we need two numbers that multiply to and sum to . These numbers are $-5$ and $2$ .

We rewrite the middle term to get $x^2 - 5x + 2x - 10 = 0$ . Grouping gives us $x(x - 5) + 2(x - 5) = 0$ , which simplifies to $(x + 2)(x - 5) = 0$ .

Equating each factor to zero, the final roots are $x = 5$ and $x =$ .

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← previousIdentifying Quadratic Equations next →Nature Of Roots & The Quadratic Formula

Concept

Roots and Zeroes

Explain roots of an equation vs zeroes of a polynomial.

A quadratic equation looks like $ax^2 + bx + c = 0$ . When we solve it, we are looking for values of $x$ that make the equation true.

If we replace $x$ with a real number $\alpha$ , and it perfectly balances the equation so that $a\alpha^2 + b\alpha + c = 0$ , we call $\alpha$ a root of the quadratic equation.

1 / 2

Method

How to Split the Middle Term

Explain the core logic of finding two numbers to split the 'bx' term.

To solve $ax^2 + bx + c = 0$ by factorisation, we split the middle term ( $bx$ ) into two parts.

But we can't just pick any two numbers that add up to $b$ . There is a strict rule: the two numbers must multiply to equal exactly $(a \times c)$ .

1 / 3

Example

Example 3: Splitting the Middle Term

Solving 2x^2 - 5x + 3 = 0.

Problem. Find the roots of the equation $2x^2 - 5x + 3 = 0$ , by factorisation.

1 / 4

Example

Example 5: Equations with Square Roots

Solving 3x^2 - 2\sqrt{6}x + 2 = 0.

Problem. Find the roots of the quadratic equation $3x^2 - 2\sqrt{6}x + 2 = 0$ .

1 / 4

reading

Your Turn: Factorisation

Faded example for solving x^2 - 3x - 10 = 0.

0 of 3 blanks filled

To solve $x^2 - 3x - 10 = 0$ , we need two numbers that multiply to and sum to . These numbers are $-5$ and $2$ .

We rewrite the middle term to get $x^2 - 5x + 2x - 10 = 0$ . Grouping gives us $x(x - 5) + 2(x - 5) = 0$ , which simplifies to $(x + 2)(x - 5) = 0$ .

Equating each factor to zero, the final roots are $x = 5$ and $x =$ .