Question

Solve the quadratic equation using the Quadratic Formula. Then solve the equation using another method. Which method do you prefer? Explain.\(2 x^2-54=12 x\)

Short answer

The roots of the equation \(2x^2 - 54 = 12x\) are \(x = 9\) and \(x = -3\). Both methods (Quadratic Formula and factoring) provide the correct roots. The preferred method would depend on the nature of the equation and personal preference.

Step-by-step solution

Organize the equation

First, arrange the given equation in the form of \( ax^2 + bx + c = 0 \). The given equation is \( 2x^2 - 54 = 12x \), which can be written as \( 2x^2 - 12x - 54 = 0 \).

Find roots using Quadratic Formula

Using the coefficients \( a = 2, b = -12, c = -54 \) into the Quadratic Formula \( x = {-b \pm \sqrt{b^2-4ac}} / {2a} \), the solutions (x-values) can be computed as \( x = {12 \pm \sqrt{(-12)^2-4(2)(-54)}} / {2(2)} \). After calculations, the roots are \( x = 9 \) and \( x = -3 \).

Solve using factoring

As an alternative method, the equation \( 2x^2 - 12x - 54 = 0 \) can be solved by factoring. After factoring, the equation can be written as \( 2(x - 9)(x + 3) = 0 \). Solving for \( x \) gives \( x = 9 \) and \( x = -3 \) as roots.

Compare and analyze methods

Both the Quadratic Formula and factoring methods provided the correct roots. The Quadratic Formula is a universal method and can solve any quadratic equation, but requires more calculations. Factoring is simpler and quicker but not all quadratic equations can be factored, especially those with irrational roots. The preferable method thus depends on the equation and personal preference.

Key concepts

Quadratic Formula

The quadratic formula is a powerful mathematical tool used to solve quadratic equations in the standard form of \( ax^2 + bx + c = 0 \). It provides a means to find the roots, or solutions, of the equation. The formula itself is expressed as:

• \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \)

Here, \( a \), \( b \), and \( c \) are constants derived from the quadratic equation. The symbol \( \pm \) indicates that there will be two solutions: one for addition and one for subtraction within the formula.
Applying the quadratic formula involves substituting the values of \( a \), \( b \), and \( c \) from your equation, calculating what's under the square root (the discriminant \( b^2 - 4ac \)), and then determining the two potential solutions. This method is valued because of its universality—it will work for any quadratic equation, regardless of whether the roots are real or complex. Nevertheless, it's worth noting that calculations can get quite intricate, especially with non-integer coefficients.

Factoring Quadratics

Factoring quadratics is an alternative method to solve quadratic equations. The trick with factoring is to express the quadratic expression as a product of two binomials. For example, \( 2x^2 - 12x - 54 = 0 \) can be factored as \( 2(x - 9)(x + 3) = 0 \).
To factor successfully, you identify two numbers that multiply to give \( ac \) (the product of \( a \) and \( c \)) and add to give \( b \), the middle coefficient.
Factoring is more intuitive and requires less calculation when you're dealing with simpler numbers. It's a "shortcut" method, much faster than the quadratic formula when applicable. However, the drawback is that not every quadratic can be easily factored, especially if the roots are irrational or complex. Factoring requires a sharp eye for recognizing number patterns, and can sometimes be impossible if the equation does not neatly break into factors.

Mathematical Methods Comparison

When comparing the quadratic formula and factoring, each method has its own clear advantages and disadvantages. The quadratic formula is very reliable—it works for any quadratic equation, no matter how complex or simple. Since it's formulaic, it eliminates the need for trial and error when an equation doesn't factor neatly.
On the flip side, factoring is preferable for equations where the coefficients are simple and factorable since it's often quicker and requires less computational work. It's also a more straightforward method for those comfortable with basic algebraic manipulation.

• If the equation has simple numbers or clearly factorable terms, factoring might be more efficient.
• If the equation seems complex or does not factor easily, the quadratic formula is more dependable.

Ultimately, the choice of method depends on the nature of the quadratic equation as well as personal preference and comfort with each technique.