Problem 65 - Algebra 2

Question

Solve each equation by completing the square. \(x^{2}+\frac{1}{3} x-\frac{35}{36}=0\)

Step-by-Step Solution

Answer

The solutions are \(x = \frac{5}{6}\) and \(x = -\frac{7}{6}\).

Step 1: Move the constant term to the other side

The given equation is \(x^{2} + \frac{1}{3}x - \frac{35}{36} = 0\). Start by moving the constant term (\(-\frac{35}{36}\)) to the other side, resulting in \(x^{2} + \frac{1}{3}x = \frac{35}{36}\).

Step 2: Find the coefficient of x and halve it

Take the coefficient of \(x\), which is \(\frac{1}{3}\), divide it by 2 to get \(\frac{1}{6}\).

Step 3: Square the result from Step 2

Square \(\frac{1}{6}\) to get \(\frac{1}{36}\). This is the number to complete the square.

Step 4: Add and subtract the same value

Add \(\frac{1}{36}\) to both sides of the equation: \[x^{2} + \frac{1}{3}x + \frac{1}{36} = \frac{35}{36} + \frac{1}{36}\].

Step 5: Simplify the right side of the equation

Simplify the right side: \(\frac{35}{36} + \frac{1}{36} = 1\). The equation becomes \(x^{2} + \frac{1}{3}x + \frac{1}{36} = 1\).

Step 6: Rewrite the left side as a perfect square

The left side \(x^{2} + \frac{1}{3}x + \frac{1}{36}\) is a perfect square trinomial and can be written as \((x + \frac{1}{6})^2 = 1\).

Step 7: Solve for x using square roots

Take the square root of both sides: \(x + \frac{1}{6} = \pm 1\). This gives two equations: \(x + \frac{1}{6} = 1\) and \(x + \frac{1}{6} = -1\).

Step 8: Solve the two linear equations

1. Solve \(x + \frac{1}{6} = 1\): Subtract \(\frac{1}{6}\) from both sides to get \(x = \frac{6}{6} - \frac{1}{6} = \frac{5}{6}\). 2. Solve \(x + \frac{1}{6} = -1\): Subtract \(\frac{1}{6}\) from both sides to get \(x = -\frac{6}{6} - \frac{1}{6} = -\frac{7}{6}\).

Key Concepts

Quadratic Equations

Quadratic equations are mathematical expressions that involve a variable raised to the second power. The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). These equations are foundational in algebra and appear in various scientific and engineering contexts.
The characteristic feature of quadratic equations is their parabolic graph, which can open upwards or downwards depending on the sign of \( a \). Finding the solutions to a quadratic equation, also known as the roots, is a critical exercise in algebra, as it helps determine the points where the graph intersects the x-axis.
Methods for solving quadratic equations include factoring, using the quadratic formula, and completing the square, which we'll delve into next.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. For quadratic equations, a variety of methods can be employed. One of the most intuitive methods is completing the square, which is particularly useful when the equation doesn't factor neatly. This method transforms the quadratic into a form where the left side is a perfect square trinomial.
Here's a quick glance at the steps involved in completing the square:

• Move the constant term to the other side of the equation.
• Take half of the coefficient of the linear term (term with \( x \)), square it, and then add and subtract this value from the same side.
• Express the modified side as a perfect square trinomial.

This process simplifies the expression, enabling you to take square roots on both sides to find the solutions for \( x \). Always remember, solutions may be real or complex, and techniques can vary based on the specific quadratic you're dealing with.

Perfect Square Trinomial

A perfect square trinomial is a quadratic expression that can be written as the square of a binomial. In general form, it's expressed as \( (x + d)^2 \), which expands to \( x^2 + 2dx + d^2 \). Creating a perfect square trinomial is a crucial part of the "completing the square" process.
In this exercise, the expression \( x^2 + \frac{1}{3}x + \frac{1}{36} \) becomes a perfect square trinomial. This transforms into \( \left(x + \frac{1}{6}\right)^2 \), simplifying the solving process considerably.
Recognizing when you're working with a perfect square trinomial is key, as it allows you to employ simpler algebraic techniques, such as leveraging square roots, to solve equations. Once the trinomial is simplified, solving for \( x \) involves easy arithmetic with linear expressions, leading quickly to the equation's solutions.