Factorise 2×2-7x-15
by splitting the middle term and by using factor
The Correct Answer and Explanation is:
To factorise the quadratic expression 2x² – 7x – 15 by splitting the middle term, we follow these steps:
Step 1: Multiply the coefficient of x² (which is 2) with the constant term (which is -15):
2 × (-15) = -30
Step 2: Find two numbers that multiply to give -30 and add up to the middle term, which is -7.
The numbers are -10 and +3 because:
-10 × 3 = -30
-10 + 3 = -7
Step 3: Split the middle term (-7x) using -10x and +3x:
2x² – 10x + 3x – 15
Step 4: Group the terms in pairs:
(2x² – 10x) + (3x – 15)
Step 5: Factor out the greatest common factor (GCF) from each group:
2x(x – 5) + 3(x – 5)
Step 6: Factor out the common binomial factor:
(x – 5)(2x + 3)
Final Answer:
(x – 5)(2x + 3)
Explanation
Factoring quadratic expressions is a fundamental skill in algebra that helps simplify expressions and solve equations. One effective method is splitting the middle term, particularly useful when the coefficient of x² is not 1. The expression 2x² – 7x – 15 fits this case because the coefficient of x² is 2.
We begin by multiplying the first and last coefficients: 2 and -15. Their product is -30. Our next goal is to find two numbers that multiply to give -30 and add up to the middle term coefficient, which is -7. After trying factor pairs of -30, we find that -10 and 3 satisfy both conditions.
Once we identify these numbers, we rewrite the middle term (-7x) as the sum of -10x and 3x. This allows us to split the expression into four terms, enabling grouping. By pairing the first two terms and the last two, we can factor each group: from the first group we factor out 2x, and from the second group we factor out 3. Both groups now contain the binomial (x – 5), which we then factor out.
The final result, (x – 5)(2x + 3), is the fully factorised form. This process simplifies the original quadratic expression and can also be used to find the roots of the corresponding equation by setting each factor to zero. Mastering this method improves understanding of polynomials, helps in solving equations, and builds algebraic fluency essential for higher mathematics.
