Final answer:
The values for a and b in the equation (x + a)² = x² + 10x + b are a = 5 and b = 25, found by expanding the binomial and equating coefficients.
Step-by-step explanation:
The student is given a binomial expansion (x + a)² and is tasked with matching it to a quadratic equation of the form x² + 10x + b. To find the values of a and b, we must expand the binomial and compare the coefficients to the given quadratic equation.
Expanding (x + a)², we get x² + 2ax + a². Now, to match the given form x² + 10x + b, the coefficient of x must be 10, and the constant term must be 'b'. Hence, we equate the coefficients: 2ax = 10x, which simplifies to 2a = 10 by dividing both sides by x, yielding a = 5. The constant term a² must then equal 'b', meaning b = a² = 5² = 25.