Problem 1 - Algebra for College Students

Question

Solve each quadratic equation by using (a) the factoring method and (b) the method of completing the square. $$x^{2}-4 x-60=0$$

Step-by-Step Solution

Answer

The solutions are \(x = 10\) and \(x = -6\).

Step 1: Identify Factors

To solve by factoring, we first need to factor the quadratic equation. We are looking for two numbers that multiply to -60 (the constant term) and add up to -4 (the coefficient of the middle term).

Step 2: Factor the Equation

The numbers -10 and 6 multiply to -60 and add up to -4. Therefore, we can factor the equation as \((x - 10)(x + 6) = 0\).

Step 3: Solve Each Factor

Setting each factor equal to zero gives us the solutions: \(x - 10 = 0\) and \(x + 6 = 0\). Solving these, we find \(x = 10\) and \(x = -6\).

Step 4: Completing the Square Setup

Start by arranging the quadratic equation in the form \(x^2 - 4x = 60\). To complete the square, take half of the coefficient of \(x\) (which is -4), divide by 2 to get -2, then square it to get 4.

Step 5: Add and Subtract the Square

Add and subtract 4 on the left side to obtain \(x^2 - 4x + 4 - 4 = 60\). Rewrite it as \((x - 2)^2 - 4 = 60\).

Step 6: Isolate the Perfect Square

Add 4 to both sides to isolate the perfect square: \((x - 2)^2 = 64\).

Step 7: Solve the Equation

Take the square root of both sides, giving \(x - 2 = \pm 8\). This leads to two equations: \(x - 2 = 8\) and \(x - 2 = -8\).

Step 8: Solve for x

Solving these gives \(x = 10\) and \(x = -6\).

Key Concepts

Factoring Method

One of the simplest ways to solve a quadratic equation is through the factoring method. This technique relies on expressing a quadratic equation in the form \( ax^2 + bx + c = 0 \) as a product of two binomials.

• We start by identifying the two numbers that multiply to the constant term \( c \) and add to the middle term coefficient \( b \). In our equation \( x^2 - 4x - 60 = 0 \), we need two numbers that multiply to \(-60\) and add up to \(-4\).

By some trial or computation, we find these numbers to be \(-10\) and \(6\). Therefore, we can express the equation as: \((x - 10)(x + 6) = 0\).This factorization shows that at least one of the terms must be zero. Thus, we set each factor equal to zero:

• \(x - 10 = 0\)
• \(x + 6 = 0\)

Solving these equations, we find the roots of the quadratic equation: \(x = 10\) and \(x = -6\).By the end of this method, these solutions can be checked by substituting them back into the original equation to verify that either solution satisfies \(x^2 - 4x - 60 = 0\). This method is efficient when the quadratic is factorable into integers.

Completing the Square

Completing the square is another algebraic technique to solve quadratic equations, often used when factoring is complex or when deriving the quadratic formula itself. This method involves transforming the equation into a perfect square trinomial.Start by rearranging the quadratic equation \( x^2 - 4x - 60 = 0 \) in the form:\( x^2 - 4x = 60 \).To form a perfect square trinomial, take half of the \( x \) coefficient, \(-4\), then divide by 2, resulting in \(-2\), and square it to get \(4\).

• Add and subtract \(4\) on the left side: \( x^2 - 4x + 4 - 4 = 60 \).
• This simplifies to \((x - 2)^2 - 4 = 60\).

Next, isolate the perfect square by adding \(4\) to both sides, yielding:\((x - 2)^2 = 64\).By taking the square root of both sides, we solve for \(x\):

• \(x - 2 = \pm 8\)

Solving these results in two separate solutions:

• \(x - 2 = 8 \Rightarrow x = 10\)
• \(x - 2 = -8 \Rightarrow x = -6\)

Completing the square provides the same solutions \(x = 10\) and \(x = -6\). This method ensures an accurate result and illuminates the symmetry within quadratic functions.

Step-by-Step Solutions

Mastering quadratic equations involves learning how to navigate through each step of different solving methods, such as factoring and completing the square. The comprehensive step-by-step approach demystifies complex mathematics:

• It involves breaking down the problems into manageable chunks.
• By following the sequence, students can better focus on the logic behind each operation.

The factoring method described begins with identifying suitable factors and efficiently resolving them to find solutions. Completing the square, meanwhile, takes us through the creative process of manipulating an equation into a perfect square trinomial. Adopting these step-by-step solutions not only provides the tools necessary for solving quadratics in exams but also develops crucial problem-solving skills. Students learning through layered steps develop a robust understanding because each technique highlights different aspects of the quadratic's behavior and nature, fostering a deeper appreciation of algebra. These solutions empower students to tackle quadratic equations with confidence and precision.