Problem 135 - Elementary Algebra

Question

In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation. Do not solve. (a) \(x^{2}-5 x-24=0\) (b)\((y+5)^{2}=12\) (c)\(14 m^{2}+3 m=11\)

Step-by-Step Solution

Answer

Equation (a): FactoringEquation (b): Square RootEquation (c): Quadratic Formula.

Step 1: Identify the Structure of Each Equation

Analyze each equation to determine its structure. Depending on whether it is a standard quadratic equation, it can be solved by factoring, square root, or the quadratic formula.

Step 2: Analyze Equation (a)

Equation (a) is given as: \(x^{2}-5 x-24=0\)This is a standard quadratic equation in the form \(ax^2 + bx + c = 0\) where a = 1, b = -5, and c = -24. It can potentially be solved by factoring since the coefficients are integers.

Step 3: Analyze Equation (b)

Equation (b) is given as: \((y+5)^{2}=12\)This is a perfect square equation. Simplify using the square root method, as it can be easily handled by taking the square root of both sides.

Step 4: Analyze Equation (c)

Equation (c) is given as: \(14m^{2}+3m=11\)Rearrange this to the standard quadratic form: \(14m^{2}+3m-11=0\)This equation does not factor easily, so the quadratic formula \(m = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\) is the most appropriate method.

Key Concepts

Factoring Method

The factoring method is a straightforward way to solve quadratic equations by expressing the quadratic equation as a product of two binomials. To use this method, the quadratic equation must be in the form \( ax^2 + bx + c = 0 \) where \(a\), \(b\), and \(c\) are constants. The main steps for factoring are:

• Find two numbers whose product is \(a \times c\) and whose sum is \(b\)
• Rewrite the middle term using these two numbers
• Factor by grouping

Let's consider an example: \(x^2 - 5x - 24 = 0\). Here, \(a = 1\), \(b = -5\), and \(c = -24\). We need to find two numbers that multiply to \(-24\) and add to \(-5\). These numbers are \(-8\) and \(+3\). Rewrite the equation: \(x^2 - 8x + 3x - 24 = 0\). Now, factor by grouping: \(x(x - 8) + 3(x - 8) = 0\). So, the equation becomes: \((x + 3)(x - 8) = 0\). Setting each factor to 0 gives: \(x + 3 = 0\) or \(x - 8 = 0\), so \(x = -3\) or \(x = 8\).
Factoring is efficient when it's possible to easily identify a pair of numbers that satisfy the necessary conditions.

Square Root Method

The square root method is useful for solving quadratic equations that can be rewritten in the form \((expression)^2 = k\). This method is highly efficient when the equation includes a perfect square or can be transformed into one. For instance, consider the equation \((y + 5)^2 = 12\). To solve this:

• Isolate the squared term if necessary
• Take the square root of both sides of the equation
• Solve for the variable

Let's solve the example: Take the square root of both sides: \((y + 5) = \pm \sqrt{12}\). Simplify this to: \(y + 5 = \pm 2 \sqrt{3}\). Then, solve for \(y\) by subtracting 5 from both sides: \(y = -5 \pm 2 \sqrt{3}\). This gives two solutions for \(y\): \(y = -5 + 2 \sqrt{3}\) and \(y = -5 - 2 \sqrt{3}\).
The square root method is particularly handy when the quadratic equation is derived from a squared binomial.

Quadratic Formula

The quadratic formula is a universal tool for solving any quadratic equation. It is derived from the standard form of a quadratic equation \( ax^2 + bx + c = 0 \): \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] This method is beneficial especially when the equation is not easily factorable. Here are the steps:

• Ensure the equation is in standard form
• Identify coefficients \(a\), \(b\), and \(c\)
• Substitute \(a\), \(b\), and \(c\) into the quadratic formula
• Simplify to find the solutions for \(x\)

Let’s solve the equation \(14m^2 + 3m = 11\) by first rewriting it as \(14m^2 + 3m - 11 = 0\). Here, \(a = 14\), \(b = 3\), and \(c = -11\). Substitute these values into the quadratic formula: \[ m = \frac{-3 \pm \sqrt{(3)^2 - 4(14)(-11)}}{2(14)} \]. This simplifies to \[ m = \frac{-3 \pm \sqrt{9 + 616}}{28} \] which further simplifies to \[ m = \frac{-3 \pm 25}{28} \]. Therefore, the solutions are \[ m = \frac{-3 + 25}{28} = \frac{22}{28} = \frac{11}{14} \] and \[ m = \frac{-3 - 25}{28} = \frac{-28}{28} = -1 \]. The quadratic formula guarantees you find all solutions for the variable, making it a reliable method for any quadratic equation.