Factorise completely: 36a² + 12ab - 15b²
Understand the Problem
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Answer
$3(6a + 5b)(2a - b)$
Answer for screen readers
$3(6a + 5b)(2a - b)$
Steps to Solve
- Factor out the Greatest Common Divisor (GCD)
The GCD of 36, 12, and 15 is 3. Factoring out 3 from the expression, we get: $3(12a^2 + 4ab - 5b^2)$
- Factor the quadratic expression inside the parentheses
We need to find two binomials of the form $(px + qy)(rx + sy)$ such that when multiplied, they give $12a^2 + 4ab - 5b^2$. We are looking for two numbers that multiply to $12 \times -5 = -60$ and add up to $4$. These numbers are $10$ and $-6$. So, we rewrite the middle term: $12a^2 + 10ab - 6ab - 5b^2$
- Factor by grouping
Group the terms in pairs: $(12a^2 + 10ab) + (-6ab - 5b^2)$ Factor out the greatest common factor from each pair: $2a(6a + 5b) - b(6a + 5b)$
- Factor out the common binomial factor
Now we can factor out the common binomial factor $(6a + 5b)$: $(6a + 5b)(2a - b)$
- Write the final factored expression
Don't forget to include the 3 we factored out initially: $3(6a + 5b)(2a - b)$
$3(6a + 5b)(2a - b)$
More Information
Factoring is the reverse process of expanding brackets. It involves breaking down an expression into a product of simpler expressions, typically binomials or polynomials of lower degree.
Tips
- Forgetting to factor out the GCD: A common mistake is to forget to factor out the greatest common divisor (GCD) at the beginning. This can lead to a more complex factoring process and a partially factored answer.
- Incorrectly identifying the factors: Choosing the wrong numbers when factoring the quadratic expression can lead to incorrect binomial factors.
- Sign errors: Making errors with the signs of the terms while factoring can change the entire solution.
- Not fully factoring the expression: One might stop after factoring out the GCD or after a partial factorization, missing the complete factorization.
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