Which quadratic equation is equivalent to (x + 2)Ā² + 5(x+2)-6=0?
A. (u + 2)Ā² + 5(u + 2) - 6 = 0 where u = (x-2)
B. u2+4+5u-6=0 where u = (x-2)
C. uĀ²+5u-6=0 where u = (x+2)
D. uĀ²+u-6=0 where u = (x+2)
Answer (2)
Substitute u = x + 2 into the given equation.
Simplify the equation to u 2 + 5 u ā 6 = 0 .
Identify the equivalent equation from the given options.
The equivalent quadratic equation is u 2 + 5 u ā 6 = 0 where u = x + 2 ā .
Explanation
Understanding the Problem We are given the equation ( x + 2 ) 2 + 5 ( x + 2 ) ā 6 = 0 and asked to find an equivalent quadratic equation from the given options. The key here is to recognize the repeating term ( x + 2 ) and make a suitable substitution to simplify the equation.
Making the Substitution Let's substitute u = x + 2 into the given equation. This gives us: u 2 + 5 u ā 6 = 0
Finding the Equivalent Equation Now, let's compare this equation with the given options. We are looking for an equation that matches u 2 + 5 u ā 6 = 0 where u = x + 2 . The option 'uĀ²+5u-6=0 where u = (x+2)' matches this exactly.
Final Answer Therefore, the equivalent quadratic equation is u 2 + 5 u ā 6 = 0 where u = x + 2 .
Examples
Substitution is a powerful tool in mathematics that allows us to simplify complex expressions. For example, imagine you are designing a bridge and need to calculate the stress on a particular beam. The stress might be described by a complicated equation, but by substituting a simpler variable for a complex term, you can make the calculations much easier. Once you've found the stress in terms of the new variable, you can substitute back to find the stress in terms of the original variables. This technique is widely used in engineering, physics, and computer science to make problems more manageable.
The equivalent quadratic equation is obtained by substituting u = ( x + 2 ) , resulting in u 2 + 5 u ā 6 = 0 . The correct answer from the options provided is C. u 2 + 5 u ā 6 = 0 where u = ( x + 2 ) .
;
