Question

Determine whether you would use factoring, square roots, or completing the square to solve the equation. Explain your reasoning. Then solve the equation. \((x+4)^2=16\)

Short answer

The equation is best solved by taking square roots. The solutions to the equation are \(x = 0\) and \(x = -8\).

Step-by-step solution

Determine the problem type

Analyze the equation. The equation is in the form \((x+c)^2 = a\), which is a perfect square. Therefore, taking the square roots both sides is a direct and effective approach.

Apply the square root method

Square root both sides of the equation to get \(x+4 = \pm \sqrt{16}\), which simplifies into \(x+4 = \pm 4\). This gives two potential solutions for 'x'.

Solve for 'x'

Subtract '4' from both sides of each equation to isolate 'x'. This results in two solutions- \(x = 4 - 4 = 0\) and \(x = -4 - 4 = -8\).

Key concepts

Factoring Quadratic Equations

When faced with a quadratic equation, factoring is often one of the first methods to consider. Factoring involves rewriting the quadratic equation as a product of two binomial expressions. This process is useful when the equation can be easily decomposed into factors. For example, if you have an equation like \(x^2 - 5x + 6 = 0\), you would look for two numbers that multiply to give the constant term — in this case, 6 — and add up to the coefficient of the middle term, which is -5. Here, the numbers would be -2 and -3, leading to factoring the quadratic as \(x-2)(x-3) = 0\).

However, sometimes factoring can be complicated or not feasible if the quadratic does not factor nicely. That is when other methods like taking square roots, completing the square, or using the quadratic formula are considered.

Taking Square Roots

Taking square roots is a straightforward technique used when the quadratic equation is already in the form of a perfect square trinomial equal to a constant, such as the provided exercise \( (x+4)^2=16 \). In such cases, you simply take the square root of both sides of the equation, leading to \(x+c = \pm \sqrt{a}\), where 'c' and 'a' are constants.

Always remember to consider both the positive and negative square roots when solving, leading to two possible solutions. As in the exercise, squaring both sides gives us \(x+4 = \pm 4\), and solving for x yields \(x = 0\) and \(x = -8\).

Completing the Square

Completing the square is a method used to solve quadratic equations that are not easily factored or when the quadratic is not a perfect square. This technique involves converting the quadratic equation into a perfect square trinomial. The process starts with moving the constant term to the opposite side of the equation from the variable terms and then adding a number to both sides of the equation to create a perfect square on one side.

For instance, for the equation \(x^2 - 6x + 5 = 0\), you would move the 5 to the other side to get \(x^2 - 6x = -5\) and then add \(\left(\frac{-6}{2}\right)^2 = 9\) to both sides to form \(x-3)^2 = 4\), which can then be solved using the square root method. This method can always be applied to any quadratic equation, making it a powerful tool in solving quadratic equations.

Perfect Square Trinomials

Perfect square trinomials are quadratic expressions that can be written as the square of a binomial, such as \(a^2+2ab+b^2\), which is the expanded form of \( (a+b)^2 \). Recognizing these patterns is key to solving quadratic equations efficiently. When you encounter an equation that already forms a perfect square trinomial such as in the exercise \( (x+4)^2=16 \), you are dealing with a much simpler situation.

This understanding lets you apply the square root method with ease, since square rooting a perfect square trinomial gives you a clear binomial expression that leads directly to the solution, as seen with \(x + 4 = \pm 4\), resulting in the solutions \(x=0\) and \(x=-8\). Recognizing perfect square trinomials can save time and effort, allowing for more direct routes to solutions.