Problem 9 - Algebra 1

Question

Solve \(x^{2}-3 x=8\) by completing the square. Solve the equation by using the quadratic formula. Which method did you find easier?

Step-by-Step Solution

Answer

The quadratic equation \(x^{2}-3 x=8\) can be solved by both methods, completing the square and using the quadratic formula, to give the same solutions, \(x = 4, x = -1\). The ease of usage among the two methods may vary depending on individual's preference.

Step 1: Solving by Completing the Square

Begin by arranging the equation. Subtract 8 on both sides to set the equation to zero: \(x^2 - 3x - 8 = 0\). Now, to complete the square, the term in the middle, -3x, is divided by 2 and its square is added and subtracted on the left-hand side. The equation becomes \((x - \frac{3}{2})^2 - \frac{25}{4} = 0 \). Then, add \( \frac{25}{4} \) on both sides of the equation to leave the square alone on one side: \( (x - \frac{3}{2})^2 = \frac{25}{4} \). Taking the square root on both sides, we get \( x - \frac{3}{2} = \pm \frac{5}{2} \). Lastly, solve for \( x \), which gives two solutions: \( x = 4, x = -1 \).

Step 2: Solving by Quadratic Formula

To solve the same equation using the quadratic formula, \(x^2 - 3x - 8 = 0\), identify the coefficients a, b, and c from the general form of the quadratic equation \( ax^2 + bx + c = 0 \). Here, \( a = 1, b = -3, c = -8 \). Then, plug these values into the quadratic formula which is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This leads us to \( x = \frac{3 \pm \sqrt{(-3)^2 - 4.1.(-8)}}{2}\). Simplifying this expression gives two solutions: \( x = 4, x = -1) \).

Step 3: Conclusion

Both methods will give the same results, but the difficulty or ease of each method can vary depending on individual's preference. If you are comfortable with factoring or guessing and checking, you might find completing the square easier. On the other hand, if you’re the type of person who prefers straightforward methods without changing the original form of the equation, you might find the quadratic formula easier.

Key Concepts

Completing the Square

Completing the square is a technique used to solve quadratic equations. It involves rearranging the equation such that one side becomes a perfect square trinomial. This makes the equation easier to solve.
To complete the square, follow these steps:

• Start with the quadratic equation in the form: \( ax^2 + bx + c = 0 \).
• Move the constant term to the other side, for example, \( x^2 - 3x = 8 \).
• Divide the coefficient of x (here, -3) by 2, and then square the result: \( \left(\frac{-3}{2}\right)^2 = \frac{9}{4} \).
• Add and subtract this square inside the equation: \( (x^2 - 3x + \frac{9}{4}) - \frac{9}{4} = 8 \).
• This turns the left side into a perfect square: \( (x - \frac{3}{2})^2 = \frac{25}{4} \).
• Take the square root of both sides: \( x - \frac{3}{2} = \pm \frac{5}{2} \).
• Solve for x to find the solutions: \( x = 4 \) and \( x = -1 \).

This process can be a bit more involved, but it's very visual and can enhance your understanding of quadratic properties.

Quadratic Formula

The quadratic formula is a universal way to find solutions for any quadratic equation \( ax^2 + bx + c = 0 \).
It works regardless of how complicated or unfactorable the equation may seem. Here is how you apply it:

• Identify coefficients a, b, and c in your equation. For example, for \( x^2 - 3x - 8 = 0 \), \( a = 1 \), \( b = -3 \), \( c = -8 \).
• Use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• Plug in the values: \( x = \frac{3 \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot (-8)}}{2} \).
• Simplify inside the square root: \( \sqrt{9 + 32} = \sqrt{41} \).
• Solve for x to find possible solutions: \( x = \frac{3 + \sqrt{41}}{2} \) and \( x = \frac{3 - \sqrt{41}}{2} \), which simplifies to \( x = 4 \) and \( x = -1 \).

The quadratic formula is straightforward and reliable, making it a preferred choice for many students when dealing with complex equations.

Quadratic Equations Solutions

Solving quadratic equations is a key skill in algebra. Quadratic equations often have two solutions and can be solved through different methods.
Here are some common ways to solve them:

• **Factoring:** This involves expressing the quadratic equation as a product of its factors. It's the simplest method, but not all quadratics are factorable over the integers.
• **Completing the Square:** As detailed before, this method turns the equation into a perfect square trinomial. It helps visualize the geometrical aspect of quadratics.
• **Quadratic Formula:** This formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) is the most versatile method. It handles any solvable quadratic equation even when factoring is off the table.

Choosing the right method often depends on the specific equation or personal preference. Practice and familiarity with each will help you decide the most efficient approach for different contexts.