Problem 53 - Algebra 2

Question

Solve each equation by completing the square. \(x^{2}-\frac{3}{2} x-\frac{23}{16}=0\)

Step-by-Step Solution

Answer

The solutions are \(x = \frac{3}{4} + \sqrt{2}\) and \(x = \frac{3}{4} - \sqrt{2}\).

Step 1: Move the constant term to the other side of the equation

Start by moving the constant term \(-\frac{23}{16}\) to the right side of the equation. This helps in isolating the terms needed for completing the square: \( x^2 - \frac{3}{2}x = \frac{23}{16} \).

Step 2: Find the number to complete the square

To complete the square, take half of the coefficient of \(x\) (which is \(-\frac{3}{2}\)), square it, and add it to both sides. Half of \(-\frac{3}{2}\) is \(-\frac{3}{4}\), and its square is \(\left(-\frac{3}{4}\right)^2 = \frac{9}{16}\).

Step 3: Add the square to both sides

Add \(\frac{9}{16}\) to both sides to form a perfect square trinomial: \(x^2 - \frac{3}{2}x + \frac{9}{16} = \frac{23}{16} + \frac{9}{16}\). This simplifies the equation to: \(x^2 - \frac{3}{2}x + \frac{9}{16} = 2 \).

Step 4: Write the left side as a square

The left side of the equation is now a perfect square trinomial, and can be written as \(\left( x - \frac{3}{4} \right)^2\). The equation becomes: \(\left( x - \frac{3}{4} \right)^2 = 2\).

Step 5: Solve for x by taking the square root of both sides

Take the square root of both sides to solve for \(x\): \( x - \frac{3}{4} = \pm\sqrt{2} \). Then, add \(\frac{3}{4}\) to both sides, resulting in: \(x = \frac{3}{4} \pm \sqrt{2}\).

Step 6: Write the final solution

The solution to the equation \(x^2 - \frac{3}{2}x - \frac{23}{16} = 0\) is \(x = \frac{3}{4} + \sqrt{2}\) and \(x = \frac{3}{4} - \sqrt{2}\).

Key Concepts

Quadratic Equations

Quadratic equations are essentially equations of the form \( ax^2 + bx + c = 0 \) where \(a\), \(b\), and \(c\) are constants. These equations create a curve on a graph called a parabola. The graph of a quadratic equation can open upwards or downwards depending on the sign of the coefficient \(a\).

The solutions to a quadratic equation are the values of \(x\) that make the equation true. These are sometimes called the roots of the equation. There are several methods to find these roots such as factoring, using the quadratic formula, or completing the square. Each method has its own advantages, and you can choose based on the problem you are working on.
When working with quadratic equations, one might encounter terms like 'discriminant'. This is part of the quadratic formula \( \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) and can tell you about the nature of the roots, whether they are real or complex, and if real, whether they are distinct or the same.

Perfect Square Trinomial

A perfect square trinomial is a specific type of quadratic expression formed when a binomial is squared. Essentially, it takes the form \((a+b)^2\), which expands to \(a^2 + 2ab + b^2\). Recognizing this form is key in the completing-the-square method.

For example, if you have \(x^2 - \frac{3}{2}x\), you can create a perfect square trinomial by determining the value that needs to be added to complete the square.

• First, take half of the linear coefficient \(-\frac{3}{2}\), which is \(-\frac{3}{4}\).
• Now square it: \((-\frac{3}{4})^2 = \frac{9}{16}\).
• Add this value to both sides of the equation to maintain equality.

This results in the expression \((x - \frac{3}{4})^2\), which is a perfect square trinomial.

Solving Equations

Solving equations involves finding all values of the variable that make the equation true. When working on quadratic equations, particularly through completing the square, one seeks to manipulate the equation such that the squared term stands alone on one side.

Steps in Solving

• Isolate the quadratic and linear terms on one side.
• Create a perfect square trinomial by adding a necessary constant to both sides.
• Write the trinomial as a square of a binomial.
• Solve the resulting equation by appropriate algebraic methods, like taking the square root.

Each step is purposefully designed to simplify the equation, making it easier to find the values of \(x\). Understanding these steps not only aids in solving quadratic equations but also strengthens your problem-solving skills for more complex equations.

Square Root Method

The square root method is an algebraic technique used to solve equations that have been transformed into perfect square binomials, such as \((x - c)^2 = k\). Once the equation is simplified to this form, solving for the unknown variable involves using the square root.

Application

• After forming a perfect square binomial, the equation \((x - \frac{3}{4})^2 = 2\) results.
• Take the square root of both sides to get \(x - \frac{3}{4} = \pm \sqrt{2}\).
• The \(\pm\) symbol signifies that there are typically two solutions from a square root equation.
• Add \(\frac{3}{4}\) to both solutions from the previous step: \(x = \frac{3}{4} + \sqrt{2}\) and \(x = \frac{3}{4} - \sqrt{2}\).

This method is especially useful when equations are neatly expressed as perfect squares, allowing for quick and accurate solutions for \(x\).