Problem 25 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Solve the equation by completing the square. $$ 4 x^{2}-13=0 $$

Step-by-Step Solution

Answer

The solutions for the equation \(4x^2 - 13 = 0\) are \(x = \frac{\sqrt{13}}{2}\) and \(x = -\frac{\sqrt{13}}{2}\).

Step 1: Write the given equation

We are given the following quadratic equation: \[ 4x^2 - 13 = 0 \]

Step 2: Divide by a

In order to rewrite the equation in terms of a perfect square trinomial, we need to divide the entire equation by the coefficient of the x^2 term, which is 4. This gives us: \[ x^2 - \frac{13}{4} = 0 \]

Step 3: Add and subtract the square of half the coefficient of the x-term

To complete the square, we take half of the coefficient of the x term, in this case, 0, square it, and add and subtract it to the equation. This will help us create a perfect square trinomial. Since there is no x term in this equation, we don't need to perform this step. Instead, we move to the next step.

Step 4: Rewrite the equation as a perfect square trinomial

We have the equation: \[ x^2 - \frac{13}{4} = 0 \] The left-hand side of the equation is already a perfect square trinomial. Now, we just need to rewrite the equation as: \[ (x - 0)^2 - \frac{13}{4} = 0 \]

Step 5: Solve for x

Now we solve the equation for x: \[ (x - 0)^2 - \frac{13}{4} = 0 \] Add \(\frac{13}{4}\) to both sides: \[ (x - 0)^2 = \frac{13}{4} \] Taking the square root of both sides: \[ x - 0 = \pm\sqrt{\frac{13}{4}} \] \[ x = \pm\frac{\sqrt{13}}{2} \] So, the solutions for the equation are \(x = \frac{\sqrt{13}}{2}\) and \(x = -\frac{\sqrt{13}}{2}\).

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation, generally expressed in the form \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \) because if \( a \) were zero, the equation would not be quadratic but linear.

In the quadratic equation, the highest power of the variable \( x \) is 2, making it distinct from other polynomial equations and giving it its characteristic parabolic shape when graphed.

Understanding a quadratic equation is crucial, as it is commonly encountered in a wide range of scientific and engineering problems. They are used to describe various phenomena such as projectile motion and optimization problems.

Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic expression obtained when a binomial is squared. It takes the form \( (x + d)^2 \) or \( (x - d)^2 \), which can be expanded to \( x^2 + 2dx + d^2 \) and \( x^2 - 2dx + d^2 \) respectively.

• This form is useful because it simplifies the process of solving equations by completing the square.
• In the given problem, the expression \((x - 0)^2\) was identified as a perfect square trinomial of \(x^2 - \frac{13}{4} = 0\).

By recognizing and rewriting an equation as a perfect square trinomial, one can efficiently solve the quadratic equation using simpler algebraic steps.

Solving Equations Step by Step

Solving equations step by step is a strategic approach to manage complex problems by breaking them down into smaller, manageable pieces. This ensures clarity and reduces the likelihood of errors.

Here's a brief guide on how such an approach works:

• Identify the type of equation: Determine whether it’s quadratic, linear, or any other type.
• Rewrite the equation: Make necessary manipulations such as dividing or factoring to bring it into a simpler form.
• Apply suitable methods: Use methods like factoring, completing the square, or the quadratic formula to find solutions.
• Verify solutions: Check the solutions in the original equation to ensure they satisfy it.

Following these steps ensures that each component is handled separately, making the resolution of the entire equation systematic and understandable.

Square Root Method

The square root method is a technique used to solve quadratic equations, primarily useful when the equation can be expressed as a perfect square trinomial. This makes it easy to isolate the variable and find solutions by taking square roots.

To use the square root method, follow these steps:

• Ensure the equation is in the form \((x - d)^2 = e\).
• Take the square root of both sides to get \(x - d = \pm\sqrt{e}\).
• Solve for \(x\) by isolating it on one side.

In the given problem, after rewriting the equation to \((x - 0)^2 = \frac{13}{4}\), the square root of both sides was taken to solve for \(x\), resulting in \(x = \pm\frac{\sqrt{13}}{2}\). This method provides a direct route to the solution by utilizing the properties of squares.