Exercises 137–139 will help you prepare for the material covered in the next section. Solve: x(x - 7) = 3.

Key Concepts

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. In the given problem, x(x - 7) = 3 can be rearranged into a standard quadratic form by moving all terms to one side, allowing us to apply methods for solving quadratics, such as factoring, completing the square, or using the quadratic formula.

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. In this case, the left side of the equation x(x - 7) can be factored, and understanding how to factor polynomials is essential for simplifying and solving quadratic equations effectively.

Zero Product Property

The Zero Product Property states that if the product of two factors equals zero, then at least one of the factors must be zero. This principle is crucial when solving quadratic equations, as it allows us to set each factor equal to zero after factoring the equation, leading to the possible solutions for x in the equation x(x - 7) - 3 = 0.