Problem 17 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The solutions are \(x = \frac{1}{2}\) and \(x = -\frac{3}{2}\).

Step 1: Move Constant to the Other Side

Start by moving the constant term to the right side of the equation. The given equation is \(x^{2} + x - \frac{3}{4} = 0\). Add \(\frac{3}{4}\) to both sides to get: \[ x^{2} + x = \frac{3}{4} \]

Step 2: Complete the Square

To complete the square, take half of the coefficient of the linear term \(x\), which is 1. Half of 1 is \(\frac{1}{2}\). Square it to get \(\left(\frac{1}{2}\right)^2 = \frac{1}{4}\). Add \(\frac{1}{4}\) to both sides of the equation:\[ x^{2} + x + \frac{1}{4} = \frac{3}{4} + \frac{1}{4} \] which simplifies to:\[ x^{2} + x + \frac{1}{4} = 1 \]

Step 3: Write as a Perfect Square

The left side of the equation \(x^{2} + x + \frac{1}{4}\) can now be written as a perfect square:\[ \left(x + \frac{1}{2}\right)^2 = 1 \]

Step 4: Solve for x

To solve for \(x\), take the square root on both sides of the equation:\[ x + \frac{1}{2} = \pm 1 \]This gives us two separate equations:1. \(x + \frac{1}{2} = 1\)2. \(x + \frac{1}{2} = -1\)

Step 5: Find the Solutions

Solve the two equations obtained:1. For \( x + \frac{1}{2} = 1 \): - Subtract \(\frac{1}{2}\) from both sides: - \(x = 1 - \frac{1}{2} = \frac{1}{2}\)2. For \( x + \frac{1}{2} = -1 \): - Subtract \(\frac{1}{2}\) from both sides: - \(x = -1 - \frac{1}{2} = -\frac{3}{2}\)

Key Concepts

Quadratic Equations

Quadratic equations are polynomials of degree two, typically expressed in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. These types of equations curve like a U-shaped parabola on a graph. Quadratic equations abound in various fields such as physics, engineering, and business, making them essential to understand.

• They might include terms like \(x^2\), which indicates that the highest power of the variable, \(x\), is 2.
• In our specific exercise example, the quadratic equation is \(x^2 + x - \frac{3}{4} = 0\), where \(a = 1\), \(b = 1\), and \(c = -\frac{3}{4}\).
• Graphically, solving a quadratic equation means finding where the parabola associated with the equation crosses the x-axis.

Understanding the nature of quadratic equations helps to apply diverse solving techniques effectively. One of these is completing the square, which transforms such an equation into a simpler, more solvable form.

Solving Equations

Solving equations is a central skill in algebra. It involves finding values of the variables that make the equation true. For quadratic equations, we often end up with two solutions owing to the presence of the squared term. In mathematics, achieving balance across both sides of the equation is key.

• Start by isolating terms involving the variable on one side and constants on the other. Our example, \(x^2 + x = \frac{3}{4}\), is the result of moving the constant term.
• Next, algebraic manipulations including addition, subtraction, and if needed, completing the square, simplify the equation further. This may yield a form that is easily factorizable or can be further simplified.
• In the example, once the square is completed, the transformation to a perfect square form \(\left(x + \frac{1}{2}\right)^2 = 1\) makes finding the solution straightforward.

Solving these transformed equations eventually yields the variable’s values, representing solution points where the equation stands true.

Mathematical Techniques

Mathematical techniques such as completing the square significantly simplify the process of solving equations, especially quadratic ones. This particular technique turns a quadratic into a perfect square trinomial, making the solve-side more manageable.
To complete the square:

• Start by determining half of the linear term's coefficient, squaring it. For \(x^2 + x\), half of 1 is \(\frac{1}{2}\), and squaring it produces \(\frac{1}{4}\).
• Add this square to both sides of the equation, adjusting the constants. Here, it converts \(x^2 + x + \frac{1}{4}\) to be equal to 1.
• This enables rewriting the quadratic as a squared binomial \((x + \frac{1}{2})^2\), which is simpler to solve for \(x\).

The next step involves taking the square root of both sides, yielding solutions for the unknown variable easily. This technique is both powerful and versatile, useful for diverse equations beyond the simple quadratic. Completing the square is not just a method but an essential skill for deeper mathematical explorations.