Problem 34 - Algebra 2

Question

Solve each equation by completing the square. \(2 x^{2}+7 x+6=0\)

Step-by-Step Solution

Answer

The solutions are \(x = -\frac{3}{2}\) and \(x = -2\).

Step 1: Move Constant to the Other Side

Start with the equation given: \(2x^2 + 7x + 6 = 0\). Move the constant term (6) to the right side to get: \(2x^2 + 7x = -6\).

Step 2: Divide by the Leading Coefficient

The equation is \(2x^2 + 7x = -6\). Divide every term by 2 to make the coefficient of \(x^2\) equal to 1: \(x^2 + \frac{7}{2}x = -3\).

Step 3: Complete the Square

To complete the square, take half of the coefficient of \(x\), square it, and add it to both sides. Half of \(\frac{7}{2}\) is \(\frac{7}{4}\). Squaring \(\frac{7}{4}\) gives \(\frac{49}{16}\). Add this to both sides: \(x^2 + \frac{7}{2}x + \frac{49}{16} = -3 + \frac{49}{16}\).

Step 4: Simplify the Equation

Simplify the right side by converting \(-3\) to a fraction with a denominator of 16: \(-3 = -\frac{48}{16}\). This results in: \(x^2 + \frac{7}{2}x + \frac{49}{16} = \frac{1}{16}\).

Step 5: Write the Perfect Square Trinomial

The left side of the equation \(x^2 + \frac{7}{2}x + \frac{49}{16}\) can be expressed as a perfect square: \((x + \frac{7}{4})^2 = \frac{1}{16}\).

Step 6: Solve for x by Taking the Square Root

Take the square root of both sides: \(x + \frac{7}{4} = \pm \frac{1}{4}\). This gives two solutions: \(x + \frac{7}{4} = \frac{1}{4}\) and \(x + \frac{7}{4} = -\frac{1}{4}\).

Step 7: Isolate x

Solve each equation for \(x\):For \(x + \frac{7}{4} = \frac{1}{4}\): Subtract \(\frac{7}{4}\) to get \(x = \frac{1}{4} - \frac{7}{4} = -\frac{6}{4} = -\frac{3}{2}\).For \(x + \frac{7}{4} = -\frac{1}{4}\): Subtract \(\frac{7}{4}\) to get \(x = -\frac{1}{4} - \frac{7}{4} = -\frac{8}{4} = -2\).

Key Concepts

Quadratic Equations

Quadratic equations are fundamental in algebra and come in the form of \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are constants, with \(a\) never being zero. The quadratic term is \(ax^2\), the linear term is \(bx\), and \(c\) is the constant. These equations are crucial as they frequently arise in various real-world scenarios, from physics to finance.
The solutions of quadratic equations are the values of \(x\) that make the equation true. Understanding how to solve these equations allows you to find these critical points.

• To solve, there are several methods available, including factoring, using the quadratic formula, graphing, and completing the square.
• Completing the square is especially beneficial when simplifying an equation to give you roots that are easier to visualize or solve directly.

A well-rounded understanding of quadratic equations will enhance problem-solving skills significantly.

Solving Equations

Solving equations, such as quadratic ones, is about finding values for variables that make the equation true. The process can vary in complexity depending on the equation's form and the technique applied. For quadratic equations, common solving methods include:
- **Factoring**: Breaking the equation down into simpler binomials if possible. This is quick but not always applicable.
- **Completing the Square**: Modifying the equation to make one side a perfect square trinomial. This is systematic and works even when factoring is difficult.
- **Quadratic Formula**: Giving direct solutions using \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This method works universally for any quadratic equation.

• Each method has its conveniences and limitations.
• Mastering more than one method provides flexibility and confidence in tackling different types of equations.

Understanding these methods will empower you to choose the most efficient one, thus saving time and effort.

Perfect Square Trinomial

A perfect square trinomial is an expression that can be written as the square of a binomial, like \((x + d)^2\). It's called "perfect" because it neatly squares into binomial form. These trinomials are crucial in solving quadratic equations by completing the square, as they simplify the process significantly.
To form a perfect square trinomial, follow these key steps:

• Divide the linear coefficient by 2.
• Square the result.
• Add this square to both sides of the equation.

For example, with an equation like \(x^2 + \frac{7}{2}x + \frac{49}{16} = \frac{1}{16}\), the trinomial \(x^2 + \frac{7}{2}x + \frac{49}{16}\) becomes \((x + \frac{7}{4})^2\).
Recognizing and forming these trinomials is essential for efficiently completing the square, bringing clarity and method to quadratic solutions.