To solve the equation 2a * 2b = -7 * -7, we need to understand the properties of roots and how they relate to quadratic equations. Let's break down the steps to solve this equation.

Understanding Quadratic Equations:
A quadratic equation is a second-degree polynomial equation in a single variable, typically written in the form ax^2 + bx + c = 0. In this case, we have a simpler form of the equation, which can be rewritten as 4ab = 49.

Solving the Equation:
To solve for the values of a and b, we need to use the properties of roots of a quadratic equation. Specifically, the sum and product of the roots.

1. Sum of the Roots:
The sum of the roots of a quadratic equation can be found using the coefficient of the linear term (b) divided by the coefficient of the quadratic term (a). In this case, since we have 2a and 2b, the sum of the roots is (2a + 2b) = 0.

2. Product of the Roots:
The product of the roots of a quadratic equation can be found using the constant term (c) divided by the coefficient of the quadratic term (a). In this case, since the constant term is -7 * -7 = 49, the product of the roots is (2a * 2b) = 49.

Equating the Sum and Product:
Now, we can equate the sum and product of the roots to solve for the values of a and b.

- Sum: 2a + 2b = 0
- Product: 2a * 2b = 49

Dividing the product equation by 4, we get ab = 49/4.

Using the sum equation, we can rewrite it as a + b = 0, which means a = -b.

Substituting -b for a in the product equation, we get (-b)b = 49/4. Simplifying, we get b^2 = 49/4.

Taking the square root of both sides, we get b = ±√(49/4).

Simplifying further, b = ±7/2.

Substituting the values of b into the sum equation, we get a + (±7/2) = 0.

Simplifying, we get a = (∓7/2).

Therefore, the solutions for a and b are a = ±7/2 and b = ±7/2.

In summary, the equation 2a * 2b = -7 * -7 can be solved using the properties of roots of a quadratic equation, specifically the sum and product of the roots. By equating the sum and product, we can solve for the values of a and b, which in this case are a = ±7/2 and b = ±7/2.