Problem 36 - Algebra for College Students

Question

Use the method of completing the square to solve each quadratic equation. $$3 x^{2}+12 x-2=0$$

Step-by-Step Solution

Answer

x = -2 ± √(14/3)

Step 1: Move Constant Term to the Other Side

To complete the square, you first need to isolate the quadratic and linear terms on one side of the equation. Begin by adding 2 to both sides of the equation to get:\[ 3x^2 + 12x = 2 \]

Step 2: Factor Out the Leading Coefficient

The quadratic term has a coefficient of 3. Factor this out from the left side of the equation:\[ 3(x^2 + 4x) = 2 \]

Step 3: Complete the Square

To complete the square, take the coefficient of the linear term (which is 4), divide it by 2, and square the result. \[ \left(\frac{4}{2}\right)^2 = 4 \]Add and subtract this square inside the parentheses:\[ 3(x^2 + 4x + 4 - 4) = 2 \]Simplify to:\[ 3((x+2)^2 - 4) = 2 \]

Step 4: Simplify the Expression

Distribute the 3 across the binomial expression:\[ 3(x+2)^2 - 12 = 2 \]

Step 5: Isolate the Square Term

Add 12 to both sides to move the constant term:\[ 3(x+2)^2 = 14 \]Then divide by 3 to solve for the square term:\[ (x+2)^2 = \frac{14}{3} \]

Step 6: Solve for x

Take the square root of both sides:\[ x+2 = \pm \sqrt{\frac{14}{3}} \]Subtract 2 from both sides to solve for \(x\):\[ x = -2 \pm \sqrt{\frac{14}{3}} \]

Key Concepts

Quadratic Equation Solving

Solving quadratic equations involves finding the values of the variable that satisfy the equation. A quadratic equation is usually in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are coefficients. There are several methods to solve these equations, including:

• Completing the square
• Using the quadratic formula
• Factoring quadratics

Each method has its own advantages depending on the specific equation. Completing the square is particularly useful when the quadratic does not factor easily or when accuracy is critical in obtaining roots. In the given exercise, completing the square is demonstrated as a method to solve \( 3x^2 + 12x - 2 = 0 \). This process involves rearranging, simplifying, and algebraically manipulating the equation to isolate and solve for \( x \).

Factoring Quadratics

Factoring quadratics is one of the simplest and most intuitive ways to solve quadratic equations. This method depends on expressing the quadratic equation in a factorable form, \( (x - p)(x - q) = 0 \), where \( p \) and \( q \) are the solutions to the equation. When the quadratic equation can be factored easily, this approach is usually the quickest.

In cases where the quadratic expression is factorable,

• Set the quadratic equation to zero
• Factor the quadratic expression
• Use the Zero Product Property to solve for \( x \)

However, not all quadratic equations are easily factorable. That's when other methods, such as completing the square or using the quadratic formula, become more advantageous.

Quadratic Formula

The quadratic formula is a universal tool used to solve any quadratic equation of the form \( ax^2 + bx + c = 0 \). It is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula provides a reliable solution for quadratics even when factoring is not possible or is complex. The expression under the square root, \( b^2 - 4ac \), is known as the discriminant, and it indicates the nature of the roots:

• If the discriminant is positive, the equation has two distinct real roots.
• If it is zero, there is exactly one real root.
• If it is negative, the equation has no real roots (the roots are complex).

Using the quadratic formula ensures precise results and is particularly helpful in equations that are difficult to factor or when completing the square becomes cumbersome.

Polynomial Equations

Polynomial equations are equations that involve terms that are powers of the unknown variable, specifically with whole number exponents. Quadratics are a type of polynomial equation where the highest degree is 2. They take the form \( ax^2 + bx + c = 0 \).

Solving polynomial equations, like quadratics, can involve different techniques. The right choice among these depends on the specific polynomial:

• Factoring: Effective for lower-degree polynomials.
• Using the quadratic formula: Suitable for quadratics regardless of complexity.
• Completing the square: Particularly helpful for equations that do not easily factor.

The goal in solving polynomial equations is to find the roots or zeros of the polynomial, which are the solutions for the equation. Understanding these solutions helps in graphing polynomial functions and analyzing their behavior.