Final answer:
The quadratic expression x^2 + 5x - 7 cannot be factored using an area model due to the lack of integer factors for -7 that add up to 5. As a result, the roots must be found using the quadratic formula and then checked for correctness.
Step-by-step explanation:
To factor the quadratic expression x^2 + 5x - 7 using an area model, we would typically look for two numbers that multiply together to give the constant term (in this case, -7), and add together to give the coefficient of the linear term (in this case, 5). However, since -7 is a prime number and there are no two integers that satisfy these conditions, factoring this expression by the area model is not possible. Therefore, we can either use the quadratic formula or complete the square to find the roots of the quadratic equation.
First, let's check if the expression could be factored as a difference of squares or a perfect square, but that is not the case here. Since the quadratic expression doesn't factor neatly, we can apply the quadratic formula x = (-b ± √(b^2 - 4ac))/(2a), with a = 1, b = 5, and c = -7, to find the roots. After substituting these values, we simplify to get the solutions for x.
As a last step, we should check the answer to ensure it is reasonable. This verification can involve substituting the solutions back into the original equation to see if they yield a true statement.