Problem 58 - Precalculus Mathematics for Calculus

Question

Solve the equation by completing the square. $$x^{2}+3 x-\frac{7}{4}=0$$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{1}{2}\) and \(x = -\frac{7}{2}\).

Step 1: Move Constant to the Other Side

Start with the equation: \[ x^2 + 3x - \frac{7}{4} = 0 \].Add \( \frac{7}{4} \) to both sides to isolate the quadratic and linear terms: \[ x^2 + 3x = \frac{7}{4} \].

Step 2: Find the Coefficient for Completing the Square

Take the coefficient of \(x\), which is 3, divide it by 2 and square it: \[ \left( \frac{3}{2} \right)^2 = \frac{9}{4} \].Add and subtract \( \frac{9}{4} \) on the left side to complete the square.

Step 3: Complete the Square

Add \( \frac{9}{4} \) to both sides to maintain equality:\[ x^2 + 3x + \frac{9}{4} = \frac{7}{4} + \frac{9}{4} \].This simplifies to:\[ x^2 + 3x + \frac{9}{4} = \frac{16}{4} = 4 \].

Step 4: Form a Perfect Square Trinomial

The left side is now a perfect square trinomial:\[ (x + \frac{3}{2})^2 = 4 \].This is because:\[ (x + \frac{3}{2})^2 = x^2 + 2 \cdot \frac{3}{2} \cdot x + \left( \frac{3}{2} \right)^2 \].

Step 5: Solve for x

Take the square root of both sides:\[ x + \frac{3}{2} = \pm 2 \].This gives two equations:\[ x + \frac{3}{2} = 2 \]and\[ x + \frac{3}{2} = -2 \].

Step 6: Solve Each Equation

Solve the first equation:\[ x + \frac{3}{2} = 2 \] by subtracting \( \frac{3}{2} \):\[ x = 2 - \frac{3}{2} = \frac{4}{2} - \frac{3}{2} = \frac{1}{2} \].Solve the second equation:\[ x + \frac{3}{2} = -2 \] by subtracting \( \frac{3}{2} \):\[ x = -2 - \frac{3}{2} = -\frac{4}{2} - \frac{3}{2} = -\frac{7}{2} \].

Key Concepts

Understanding Quadratic Equations

Quadratic equations are expressions that involve the square of an unknown variable, typically written in the form of \( ax^2 + bx + c = 0 \). They can model various real-world scenarios, from projectile motion to finding the area of a space. The term "quadratic" comes from "quad," meaning square, as the highest degree is a square term. Here are the essential components:

• \( a \) is the coefficient of \( x^2 \),
• \( b \) is the coefficient of \( x \),
• \( c \) is the constant term.

In our example, the quadratic equation is \( x^2 + 3x - \frac{7}{4} = 0 \). Here, \( a = 1 \), \( b = 3 \), and \( c = -\frac{7}{4} \). To solve this equation, we can use a method called "completing the square," which transforms the quadratic into a perfect square trinomial. This approach simplifies finding the values of \( x \) that satisfy the equation.

Crafting a Perfect Square Trinomial

A perfect square trinomial is a special quadratic expression that can be expressed as the square of a binomial. This means it has the form \( (x + d)^2 \). The concept of completing the square is about reshaping the quadratic equation to fit this form.
Here's how to identify a perfect square trinomial:

• The first term \( x^2 \) and the third term \( c \) must be perfect squares.
• Twice the product of \( x \) and the square root of \( c \) should equal the middle term \( bx \).

In the given problem, after isolating the quadratic terms, we use the coefficient of \( x \) to complete the square.- Divide \( b \) by 2 (\( \frac{3}{2} \)) and square it (\( \frac{9}{4} \)).- Add and subtract this value to transform the equation into a perfect square trinomial: \( x^2 + 3x + \frac{9}{4} \).- This gives \( (x + \frac{3}{2})^2 \). So, the expression is perfectly squared, leading to a simpler solution process.

Solving Equations Using the Square Root Approach

Once the equation is transformed into a perfect square trinomial, solving for the unknown variable becomes straightforward using the square root method. This means we take the square root of both sides, remembering to include both positive and negative solutions. Let's outline the steps:

• From \( (x + \frac{3}{2})^2 = 4 \), take the square root of both sides.
• We obtain \( x + \frac{3}{2} = \pm 2 \), which leads to two potential solutions:
• \( x + \frac{3}{2} = 2 \)
• \( x + \frac{3}{2} = -2 \)

Next, solve each resulting equation:- For \( x + \frac{3}{2} = 2 \), subtract \( \frac{3}{2} \) to find \( x = \frac{1}{2} \).- For \( x + \frac{3}{2} = -2 \), subtract \( \frac{3}{2} \) to find \( x = -\frac{7}{2} \).Through completing the square and using the square root method, we found both possible values for \( x \), completing the solution of the quadratic equation.