Solve each equation in Exercises 83–108 by the method of your choice. 4x^2 - 16 = 0

Verified step by step guidance

1. Start by observing the equation: $4x^2 - 16 = 0$. Notice that this is a quadratic equation in the form $ax^2 + bx + c = 0$, where $a = 4$, $b = 0$, and $c = -16$.
2. To simplify the equation, add 16 to both sides to isolate the $x^2$ term: $4x^2 = 16$.
3. Next, divide both sides of the equation by 4 to solve for $x^2$: $x^2 = 4$.
4. Now, take the square root of both sides to solve for $x$. Remember that taking the square root introduces both positive and negative solutions: $x = \pm \sqrt{4}$.
5. Simplify the square root to find the solutions for $x$: $x = \pm 2$.

Key Concepts

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to these equations can be found using various methods, including factoring, completing the square, or applying the quadratic formula. Understanding the structure of quadratic equations is essential for solving them effectively.

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. For quadratic equations, this often involves expressing the equation in the form (px + q)(rx + s) = 0. This method is particularly useful when the quadratic can be easily factored, allowing for straightforward identification of the roots.

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 factored equations, as it allows us to set each factor equal to zero and solve for the variable. Applying this property is a key step in finding the solutions to quadratic equations after factoring.