Problem 37 - Precalculus : Building Concepts and Connections

Question

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

Step-by-Step Solution

Answer

The solutions to the quadratic equation are \(x = 1\) and \(x = -1.5\).

Step 1: Rearrange the Equation

The equation is already in the form \(x^{2}+x=2\), so there's no need to rearrange it.

Step 2: Complete the Square

The coefficient of \(x\) is 1, so \(b = 1\). Therefore, we add \((b/2)^2 = (1/2)^2 = 0.25\) to both sides of the equation to form a perfect square. The equation becomes \(x^2 + x + 0.25 = 2 + 0.25\).

Step 3: Simplify the Equation

Simplify the equation to get \(x^2 + x + 0.25 = 2.25\). Now, the equation is in the form of a perfect square trinomial.

Step 4: Factor the Trinomial

Factor the left side of the equation, which simplifies the equation to \((x + 0.5)^2 = 2.25\).

Step 5: Apply the Square Root Property

Apply the square root property, which states that if \(x^2 = c\), then \(x = \sqrt{c}\) or \(x = -\sqrt{c}\). The equation becomes \(x + 0.5 = ± \sqrt{2.25}\).

Step 6: Solve for \(x\)

Solving for \(x\), \(x = -0.5 ± \sqrt{2.25}\) which gives \(x = 1\) or \(x = -1.5\).

Key Concepts

Understanding Quadratic Equations

Quadratic equations are a fundamental aspect of algebra. They are equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents the variable. The highest power of \(x\) is 2, which is why it’s called "quadratic," derived from the Latin word "quadratus," meaning "square."

• The standard form of a quadratic equation is \(ax^2 + bx + c = 0\).
• The solutions to quadratic equations are known as "roots," and they can be real or complex numbers.
• Quadratic equations can be solved using various methods, such as factoring, the quadratic formula, completing the square, or graphing.

Understanding quadratic equations is crucial for solving complex algebra problems and for applications in various fields such as physics, engineering, and economics.Completing the square is specifically useful for solving quadratic equations when factoring is insufficient or the quadratic formula seems cumbersome. This method transforms the equation into a perfect square trinomial, making it easier to solve.

Factoring Trinomials in Quadratic Equations

Factoring trinomials is an essential technique when unraveling quadratic equations. A trinomial, like \(x^2 + x + 0.25\) from our given example, can often be rewritten as the product of two binomials. When solving a quadratic equation by completing the square, we aim to restructure it into a perfect square trinomial.
Factoring plays a significant role because:

• It simplifies the equation, making it easier to apply the square root property and find solutions.
• A trinomial that can be expressed as a perfect square is easier to manage because it has a clear structure, \((x + m)^2\), leading directly to solutions.
• This method shows the relationships among terms, especially useful when other solution methods might be too complex or when dealing with irrational numbers.

By turning trinomials into recognizable patterns, you not only simplify your calculations but also deepen your understanding of how components within equations interact.

Using the Square Root Property

Once you have a quadratic equation in the clean form of a perfect square, \((x + m)^2 = c\), the square root property is your next step. This property states that if \(x^2 = c\), then \(x = \sqrt{c}\) or \(x = -\sqrt{c}\). This principle helps solve the equation by extracting the exact values for \(x\).
Applying the square root property is beneficial because it:

• Simplifies the equation-solving process by reducing the equation to basic roots, eliminating the complexity of multiple terms.
• Enables direct computation of the solution, bypassing more involved methods like the quadratic formula.
• Allows for the determination of all solutions, both positive and negative, which is essential for finding all roots of the equation.

In our exercise, after rewriting \(x^2 + x + 0.25 = 2.25\) as \((x + 0.5)^2 = 2.25\), the square root property helps us quickly find \(x = 1\) or \(x = -1.5\). This shows how crucial the square root property is in making calculations efficient and straightforward.