Problem 39 - Precalculus : Building Concepts and Connections

Question

Solve the quadratic equation by completing the square. $$2 x^{2}+8 x-1=0$$

Step-by-Step Solution

Answer

The solutions of the equation are \(x = -2 + \sqrt{17} \) and \(x = -2 - \sqrt{17}\)

Step 1: Rearrange the equation

First, we move the constant term to the right side of the equation. After doing so, the equation is \(2x^2 + 8x = 1\)

Step 2: Complete the square

Complete the square on the left by adding the square of half the coefficient of \(x\) to both sides. To find this value, we divide the coefficient of \(x\) by 2, then square it. Thus, \( (8/2)^2 = 16 \). After adding 16 to both sides, the equation becomes \( 2x^2 + 8x + 16 = 1 + 16 \), which simplifies to \( 2x^2 + 8x + 16 = 17 \). This equation can be rewritten as \( 2(x^2 + 4x + 8) = 17 \).

Step 3: Solve for \(x\)

Finally, dive both sides of the equation by 2 to isolate (x+2)^2 on one side, yields \( (x+2)^2 = 17 \). Taking square root on both sides; we get \(x+2 = ±\sqrt{17}\). Then, subtract 2 from both side, we get \(x = -2 ± \sqrt{17}\)

Key Concepts

Quadratic Equation

The quadratic equation is a fundamental type of polynomial equation of the second degree, which means it contains at least one term that is squared, or raised to the power of two. It takes the general form of \[ ax^2 + bx + c = 0 \], where \(a\), \(b\), and \(c\) are constants, and \(a\) is non-zero. These equations are ubiquitous in algebra and appear in various areas of mathematics and science.
Quadratic equations can have two solutions, one solution, or no real solution depending on the discriminant, which is calculated as \(b^2 - 4ac\). The solutions are found by various methods, including factoring, using the quadratic formula, graphing, and completing the square which transforms the equation into a perfect square trinomial—making it easier to solve.

Solve Quadratic Equation

To solve a quadratic equation means to find the values of \(x\) that make the equation true. The process of solving can vary based on the form of the equation and the methods applied. One common method, as demonstrated in the provided exercise, is to complete the square which involves creating a perfect square trinomial from the quadratic equation. Once done, it allows the equation to be expressed in the form of \((x+p)^2 = q\), which can be solved by taking the square root of both sides. The result yields two solutions because both positive and negative square roots must be considered. This method often simplifies the problem and is especially useful when the quadratic equation does not factorize easily.

Algebraic Methods

Algebraic methods encompass a variety of techniques used to solve equations, including those used to solve quadratic equations. When it comes to quadratics, three primary algebraic methods are factoring, using the quadratic formula, and completing the square. Factoring involves rewriting the equation in a product form, allowing for the application of the zero product property. The quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), gives the solutions directly from the coefficients of the equation without the need to manipulate its form. Completing the square, which we focus on in this instance, involves adding a constant to both sides of the equation to form a perfect square trinomial on one side.

Precalculus

Precalculus is a course that prepares students for calculus, and it covers a wide range of topics, including algebraic concepts such as the solving of quadratic equations. Understanding how to manipulate and solve quadratic equations is a vital precalculus skill. In precalculus, students learn about the behaviors of polynomial functions, applications of algebra to geometry, and various techniques for solving different types of equations. Completing the square is a technique that exemplifies the kind of algebraic manipulation integral to precalculus, providing a foundation for more advanced studies in mathematics.