Question

Solve the quadratic equation using the Quadratic Formula. Then solve the equation using another method. Which method do you prefer? Explain.\(9 x^2+36 x+72=0\)

Short answer

The roots obtained from both methods are \(x= -2\). It would be reasonable to prefer factoring for this specific equation due to its simplicity, but it greatly depends on the specific quadratic equation being dealt with.

Step-by-step solution

Solve by Quadratic Formula

The Quadratic Formula is given as \[x = {-b \pm \sqrt{b^2 - 4ac}}/{2a}\] Substitute values a=9, b=36 and c=72 into the formula. We get \[x = {-36 \pm \sqrt{36^2 - 4*9*72}}/{2*9}\]. After simplification, we obtain \(x= -2\).

Solve by Factoring

Firstly, divide all terms of the equation by 9 to simplify, we get \(x^2 + 4x + 8 = 0\). Then, decompose \(4x\) into two terms whose product is \(8\) and sum is \(4\), we get \((x+2)^2 = 0\). Solving for \(x\), we again get \(x= -2\)

Comparison of Methods

The Quadratic Formula provides a general solution for any quadratic equation, but may involve complex computation especially when numbers are large or not perfect squares. Factoring, on the other hand, is quicker and simpler when numbers are small or can be easily factored, as in this case. So depending on the specific equation, one might prefer one method over the other

Key concepts

Factoring

Factoring involves rewriting a quadratic equation as a product of two simpler binomial expressions. This is particularly useful when the quadratic is easily decomposable into whole numbers that add to the middle term and multiply to the constant term.

For example, in the quadratic equation \(x^2 + 4x + 8 = 0\), factoring means finding two numbers that multiply to 8 and add to 4.

• The pair of numbers (2, 2) satisfies both conditions: 2 + 2 = 4 and 2 \( \times \) 2 = 8.

So, we can express the quadratic as \((x + 2)^2 = 0\).

Solving \((x + 2)^2 = 0\) yields \(x = -2\). Factoring works well when the quadratic can be expressed in simple integer terms. It is a quick, elegant method but requires the equation to be set up favorably.

Completing the square

Completing the square transforms a quadratic equation into a perfect square trinomial. This method is powerful as it always works for any quadratic, providing a clear path to the solution.

To complete the square for an equation like \(x^2 + 4x + 8 = 0\), start by isolating the constant term from the squared term:

• Subtract 8 from both sides: \(x^2 + 4x = -8\).
• Take half of the coefficient of \(x\), square it, and add it to both sides. Here, \((4/2)^2 = 4\).

Adding 4 to both sides makes the equation \((x + 2)^2 = -4\). Recognizing this leads to adjusting and further solving. This method ensures the equation is solvable even when factoring is not apparent, especially with non-integer results.

Quadratic equation

A quadratic equation is any polynomial equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The highest power of the variable \(x\) is 2, giving this equation its name.

Quadratic equations appear in various scenarios, from projectile motion to finding maximal areas.

• They may have real and distinct roots, real and repeated roots, or complex roots, depending on the discriminant \(b^2 - 4ac\):
• If \(b^2 - 4ac > 0\), there are two distinct real roots.
• If \(b^2 - 4ac = 0\), there's one repeated real root.
• If \(b^2 - 4ac < 0\), the solutions are complex.

These outcomes play a significant role in determining the number and type of solutions a quadratic provides.

Methods of solving quadratic equations

There are several methods to solve quadratic equations, each with their strengths.

Let's look at the most prominent ones:

• **Factoring:** Best used when the quadratic can be easily rewritten as a product of binomials, providing a fast solution.
• **Completing the square:** A universally applicable method, transforming the quadratic into a perfect square to solve easily.
• **Quadratic Formula:** Provides solutions for any quadratic equation. It's robust but can be computationally intensive for complex numbers.
• **Graphing:** Visual method, plotting the equation to find intersections that equate to solutions. More conceptual and less precise for exact answers.

Different equations lend themselves more readily to particular methods based on their structure. Understanding the equations' characteristics and these strategies allows one to choose the best method efficiently.