Final answer:
The term needed to add to the equation x² + 10x = 9 to make it a perfect square is the square of half the coefficient of the x term, which is 25. This transforms the equation into a perfect square trinomial, (x + 5)² = 34.
Step-by-step explanation:
To find the term that must be added to the equation x² + 10x = 9 to make it a perfect square, we use the process known as completing the square. Completing the square involves creating a trinomial square from the quadratic part of the equation, which can then be factored into the form (x + a)². We do this by adding the square of half the coefficient of the x term to both sides of the equation.
In this case, the coefficient of x is 10. Half of 10 is 5, and the square of 5 is 25. So, we add 25 to both sides of the equation.
This gives us:
- x² + 10x + 25 = 9 + 25
- (x + 5)² = 34
The equation is now a perfect square trinomial, and the correct answer is 25, which is option (a).