Question

Solve the quadratic equation using the Quadratic Formula. Then solve the equation using another method. Which method do you prefer? Explain.\(5 x^2-50 x=-135\)

Short answer

The solutions to the equation using both the Quadratic Formula and factoring are x=9 and x=3. After comparing the two methods, it can be concluded that while the Quadratic Formula is more universally applicable, factoring is quicker and simpler for this specific equation.

Step-by-step solution

Solving using the Quadratic Formula

First, rearrange the equation to standard form \(ax^2 + bx + c = 0\). In this case, it becomes \(5x^2-50x+135=0\). Now, apply the Quadratic Formula which is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). For the equation \(5x^2-50x+135=0\), a=5, b=-50, and c=135. Plug these into the formula to determine the roots of the equation.

Solving using factoring

To solve the equation using factoring, rearrange the equation to the form \(ax^2 + bx + c = 0\), as before which becomes \(5x^2-50x+135=0\). This equation can be factored to \((5x-45)(x-3)=0\). The solutions are obtained by setting each factor equal to zero.

Comparative analysis

To decide which method to prefer, consider factors like complexity, time consumption and suitability for the equation in question. Factoring is usually quicker and simpler, but it can be very difficult to spot the factors for complex equations. The Quadratic Formula, on the other hand, works for any quadratic equation, but it involves multiple computational steps that can be prone to arithmetic errors. In this case, considering the simplicity and speed, factoring may be the preferred method.

Key concepts

Quadratic Formula

The Quadratic Formula is a powerful tool used to find the roots of any quadratic equation, which is in the standard form \(ax^2 + bx + c = 0\). The formula itself is expressed as: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula provides a reliable method to solve even the toughest quadratic equations. Using it, you can find the roots by simply plugging in the values of \(a\), \(b\), and \(c\) from your equation.

• a, b, c: These are coefficients of your quadratic equation. For each equation, it’s crucial to identify these values correctly.
• Discriminant (\(b^2 - 4ac\)): This part of the formula helps to determine the nature of the roots.

Some reasons why the Quadratic Formula is favored include:

• Universality: It works for all quadratic equations, regardless of whether they can be easily factored.
• Predictability: Provides precise results through consistent steps.

However, due to its complexity, it might be prone to mistakes if the arithmetic is not carefully handled. To minimize errors, take your time with each part of the calculation.

Factoring Quadratics

Factoring Quadratics is another effective method to find solutions to quadratic equations. It involves expressing the quadratic equation in a product of binomials such that it equals zero:Example: Convert \(5x^2 - 50x + 135 = 0\) into a factored form like \((5x - 45)(x - 3) = 0\).

• Set each factor equal to zero: One major step in factoring is setting each parenthesis to zero, yielding the potential solutions.
• Check for GCF (Greatest Common Factor): Before factoring, always check if there is a common factor among the terms to simplify the equation.

Generally, factoring is preferred because:

• Efficiency: It can be quicker than other methods when factors are easily identified.
• Simplicity: Fewer arithmetic steps compared to the Quadratic Formula.

A limitation is that it's not always applicable, particularly when the quadratic doesn't factor neatly. In such cases, other methods like the Quadratic Formula or Completing the Square might need to be used.

Completing the Square

Completing the Square is a third method to solve quadratic equations, particularly helpful when the equation doesn't factor easily. This method converts the quadratic into a perfect square trinomial, allowing easier manipulation.To "complete the square":

• Rearrange: Start by moving the constant term to the other side of the equation.
• Find the square: Determine the number to add to both sides of the equation that makes a perfect square trinomial.
• Solve: Simplify and solve for \(x\).

Why use Completing the Square?

• Useful for non-factorable equations: When factoring isn't feasible, this method provides a pathway to find roots.
• Leads to Quadratic Formula: This process is foundational in deriving the Quadratic Formula itself.

While detailed, it provides deep insights into the structure of quadratic equations, teaching more about their behavior and solutions over merely finding the roots.