Factorising Quadratic Equations

Study Notes

Factorising Quadratic Equations

Quadratic equations are expressed in the standard form ( ax^2 + bx + c = 0 ).
Factorisation converts a quadratic into the product of two binomials.

Steps to Factorise Quadratic Equations

Identify coefficients: Recognize ( a ), ( b ), and ( c ) from the equation.
Multiply ( a ) and ( c ) to find ( ac ).
Find two numbers that:
- Multiply to ( ac )
- Sum to ( b )
Rewrite the middle term ( bx ) using the two identified numbers.
Group terms for factoring by common factors, resulting in two binomials.

Example

For ( 2x^2 + 5x + 3 = 0 ):
- Coefficients: ( a = 2 ), ( b = 5 ), ( c = 3 )
- Product: ( ac = 6 ) (where ( 2 \cdot 3 = 6 ))
- Key numbers: ( 2 ) and ( 3 ) (since ( 2 \times 3 = 6 ) and ( 2 + 3 = 5 ))
- Rewrite as ( 2x^2 + 2x + 3x + 3 )
- Group: ( (2x^2 + 2x) + (3x + 3) )
- Factor out: ( 2x(x + 1) + 3(x + 1) )
- Final factorisation yields ( (2x + 3)(x + 1) = 0 )

Solving Quadratic Equations

After factorisation into ( (px + q)(rx + s) = 0 ):
- Set each factor to zero: ( px + q = 0 ) and ( rx + s = 0 )
- Solve for ( x ) using:
  - ( x = -\frac{q}{p} ) for ( px + q = 0 )
  - ( x = -\frac{s}{r} ) for ( rx + s = 0 )

Example of Solving

From ( (2x + 3)(x + 1) = 0 ):
- For ( 2x + 3 = 0 ): ( x = -\frac{3}{2} )
- For ( x + 1 = 0 ): ( x = -1 )
Solutions to the equation are ( x = -\frac{3}{2} ) and ( x = -1 ).

Key Points

Start by identifying the coefficients ( a ), ( b ), and ( c ).
Ensuring accurate factor pairs is crucial for correct factorisation.
Solutions obtained yield the roots of the quadratic equation, revealing where the quadratic intersects the x-axis.

Description

This quiz covers the process of factorising quadratic equations of the form ax² + bx + c = 0. Learn how to identify coefficients, multiply, and find the necessary numbers to express the quadratic as a product of binomials. Test your understanding with examples and practice problems.