Problem 4 - Algebra for College Students

Question

Solve each quadratic equation by using (a) the factoring method and (b) the method of completing the square. $$x^{2}-18 x=-72$$

Step-by-Step Solution

Answer

The solutions are \(x = 6\) and \(x = 12\).

Step 1: Bring equation to standard form

The given equation is \(x^2 - 18x = -72\). First, move 72 to the left side to form a standard quadratic equation: \(x^2 - 18x + 72 = 0\).

Step 2: Factoring the quadratic equation

To factor the equation \(x^2 - 18x + 72 = 0\), find two numbers that multiply to 72 and add to -18. The numbers -6 and -12 satisfy this condition: \((x - 6)(x - 12) = 0\).

Step 3: Solve the factored equation

Set each factor equal to zero: \(x - 6 = 0\) and \(x - 12 = 0\). Solving these gives \(x = 6\) and \(x = 12\).

Step 4: Completing the square—rearrange and halve the middle coefficient

Start with \(x^2 - 18x = -72\). To complete the square, rearrange to \(x^2 - 18x = c\). Calculate \((-18/2)^2 = 81\) and add it to both sides: \(x^2 - 18x + 81 = 9\).

Step 5: Express as a perfect square trinomial

The left side becomes a perfect square: \((x - 9)^2 = 9\).

Step 6: Solve by taking the square root

Take the square root of both sides: \(x - 9 = \pm 3\). This gives two equations: \(x - 9 = 3\) and \(x - 9 = -3\).

Step 7: Solve for x

Solving the two equations: \(x = 12\) and \(x = 6\).

Key Concepts

Factoring Method

The factoring method is a simple and efficient technique for solving quadratic equations. This method involves expressing the quadratic equation in the form of a product of two lininear binomials. For the original equation \(x^2 - 18x + 72 = 0\), the goal is to find two numbers that multiply to 72 (the constant term) and add to -18 (the coefficient of the linear term).

• Identify factors: Begin by listing pairs of numbers whose product is 72. For example, 1 and 72, 2 and 36, 3 and 24, and so on.
• Select suitable factors: Look for the pair that adds up to -18. The correct pair is -6 and -12, as they multiply to 72 and add to -18.
• Rewrite the quadratic: This allows us to rewrite the quadratic as \((x - 6)(x - 12) = 0\).

After factoring, solve each factor set to zero, leading to two potential solutions for \(x\). Thus, \(x - 6 = 0\) gives \(x = 6\), and \(x - 12 = 0\) provides \(x = 12\). The factoring method is straightforward if the quadratic can be factored easily.

Completing the Square

Completing the square offers another way to solve quadratic equations. It transforms the quadratic into a perfect square form, which makes solving for \(x\) more manageable.
To complete the square for the quadratic equation \(x^2 - 18x = -72\), the first step is to take the coefficient of \(x\), divide it by two, and then square it:

• Determine the number to add and subtract: Take \(-18\), halve it to get \(-9\), then square \(-9\) to get 81.
• Adjust the equation: Add 81 to both sides to keep the equation balanced: \(x^2 - 18x + 81 = 9\).

This transformation allows the left side of the equation to be expressed as a perfect square: \((x - 9)^2 = 9\). From this form, you can easily solve by taking the square root of each side, resulting in two linear equations: \(x - 9 = 3\) and \(x - 9 = -3\). Consequently, solving these equations gives us solutions \(x = 12\) and \(x = 6\). Completing the square is efficient and provides a clear path to the solution.

Standard Form of Quadratics

The standard form of a quadratic equation is \[ax^2 + bx + c = 0\]. This form is crucial as it sets the foundation for both the factoring method and completing the square. It organizes the terms in a way that is most useful for solving the equations. For our original problem, the equation \(x^2 - 18x = -72\) was first transformed to the standard form \(x^2 - 18x + 72 = 0\).

• General form: Every quadratic equation can be rearranged into the standard form.
• Consistency: The standard form ensures you're set up to use various solving methods efficiently.
• Identify coefficients: From \(x^2 - 18x + 72 = 0\), here \(a = 1\), \(b = -18\), and \(c = 72\).

Once we have the standard form, we know the layout and can choose the most effective method for solving, depending on the given constants. Quadratic equations take different forms in real scenarios, but converting them to the standard form is often the first step.