Quadratic Equation

Solving by Factorisation

Solve the equation:

2x² + x − 3 = 0

[3]

Step 1: Find the product of the first and third terms

Find the product of the coefficient of the first term and the constant term:

(2)(−3) = −6

Step 2: Find the two factors

Since −6 is negative, find two factors of −6 whose product is −6 and whose sum is the middle-term coefficient, +1.

The two factors must have different signs because −6 is negative. The larger factor must have the same sign as the middle term, which is positive.

These factors are:

+3 and −2

3 × (−2) = −6
3 + (−2) = 1

Step 3: Replace the middle term

Replace +x with +3x − 2x.

2x² + x − 3 = 0

2x² + 3x − 2x − 3 = 0

Step 4: Factorise by grouping

2x² + 3x − 2x − 3 = 0

x(2x + 3) − 1(2x + 3) = 0

(x − 1)(2x + 3) = 0

Step 5: Solve for x

(x − 1)(2x + 3) = 0

Therefore,

x − 1 = 0   or   2x + 3 = 0

x = 1   or   x = −3/2

Final Answer

x = 1 or x = −3/2