Solving Quadratics using Factorisation - Corbettmaths

Introduction

This tutorial will guide you through solving quadratic equations using factorization, a straightforward and efficient method. Factorization is often quicker than other methods like the quadratic formula or completing the square, making it an essential skill for solving quadratics.

Step 1: Understanding Factorization

Before diving into solving quadratics, ensure you are comfortable with the concept of factorization. Factorization involves breaking down an expression into simpler components (factors) that, when multiplied together, yield the original expression.

• Review how to factorize a quadratic expression of the form ( ax^2 + bx + c ).
• Identify two numbers that multiply to give ( ac ) (the product of the coefficient of ( x^2 ) and the constant term) and add to give ( b ) (the coefficient of ( x )).

Step 2: Setting Up the Equation

To solve a quadratic equation, ensure it is set to zero. For example, for the equation:

[ x^2 + 7x + 10 = 0 ]

• This equation is already set to zero, which is necessary for factorization.

Step 3: Factorizing the Quadratic

Start by placing two brackets in the standard format:

[ (x + p)(x + q) = 0 ]

Where ( p ) and ( q ) are the numbers you need to find.

Example 1: Factorizing ( x^2 + 7x + 10 )

1. Identify two numbers that multiply to 10 and add to 7: they are 5 and 2.
2. Write the factored form:

[ (x + 5)(x + 2) = 0 ]

Step 4: Solving the Factored Equation

Use the zero product property, which states if ( ab = 0 ), then ( a = 0 ) or ( b = 0 ).

Continuing the Example

• Set each bracket to zero:

1. ( x + 5 = 0 ) leads to ( x = -5 )
2. ( x + 2 = 0 ) leads to ( x = -2 )

Thus, the solutions are ( x = -5 ) and ( x = -2 ).

Step 5: Solving More Complex Quadratics

For equations that involve different coefficients, such as:

[ 6x^2 + 13x - 5 = 0 ]

1. Factor: Find two numbers that multiply to ( -30 ) (6 times -5) and add to 13. The numbers are 15 and -2.
2. Rewrite the equation in factored form:

[ (3x - 1)(2x + 5) = 0 ]

Step 6: Finding the Solutions

Again, apply the zero product property:

1. ( 3x - 1 = 0 ) gives ( x = \frac{1}{3} )
2. ( 2x + 5 = 0 ) gives ( x = -\frac{5}{2} )

The solutions are ( x = \frac{1}{3} ) and ( x = -\frac{5}{2} ).

Step 7: Handling Special Cases

Sometimes, the quadratic might have a repeated root or need rearranging. For example, transform:

[ x^2 - 4x + 16 = 0 ]

1. Rearrange to get it in the standard form.
2. Factor as:

[ (x - 4)(x - 4) = 0 ]

3. This shows a repeated root: ( x = 4 ).

Conclusion

You have learned how to solve quadratic equations using factorization. The key steps involve ensuring the equation is set to zero, factorizing it into brackets, and applying the zero product property to find the solutions.

Next, practice with various quadratic equations to reinforce your understanding of this method. Factorization not only simplifies solving quadratics but also enhances your algebraic skills.