Fill in the gap to factorise this expression: [tex]10 x+15=-(2 x+3)[/tex]
Answer (1)
Expand the right side of the equation: 10 x + 15 = − 2 x − 3 .
Move all terms to the left side: 10 x + 15 + 2 x + 3 = 0 .
Combine like terms: 12 x + 18 = 0 .
Factor out the greatest common divisor: 6 ( 2 x + 3 ) = 0 . The factorised expression is 6 ( 2 x + 3 ) = 0 .
Explanation
Understanding the Problem We are given the equation 10 x + 15 = − ( 2 x + 3 ) . Our goal is to factorise an expression derived from this equation.
Expanding the Right Side First, let's expand the right side of the equation: 10 x + 15 = − 2 x − 3 .
Moving Terms to One Side Next, we move all terms to the left side of the equation by adding 2 x and 3 to both sides: 10 x + 15 + 2 x + 3 = 0 .
Combining Like Terms Now, we combine like terms: ( 10 x + 2 x ) + ( 15 + 3 ) = 12 x + 18 = 0 .
Finding the Greatest Common Divisor We want to factor the expression 12 x + 18 . We find the greatest common divisor (GCD) of the coefficients 12 and 18. The GCD of 12 and 18 is 6.
Factoring out the GCD We factor out the GCD, 6, from the expression: 6 ( 2 x + 3 ) = 0 . Thus, the factorised expression is 6 ( 2 x + 3 ) = 0 .
Final Factorisation Therefore, the expression 10 x + 15 = − ( 2 x + 3 ) can be rewritten and factorised as 6 ( 2 x + 3 ) = 0 .
Examples
Factoring expressions is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to simplify complex equations when designing structures, ensuring stability and efficiency. Similarly, economists use factoring to analyze market trends and predict economic behavior. In computer science, factoring is used in cryptography to secure data and protect against cyber threats. Understanding factoring helps in simplifying problems and finding efficient solutions across various fields.
