Problem 7 - Algebra for College Students

Question

Solve each quadratic equation by using (a) the factoring method and (b) the method of completing the square. $$x(x+7)=8$$

Step-by-Step Solution

Answer

The solutions are \(x = -8\) and \(x = 1\).

Step 1: Expand the equation

The given quadratic equation is \(x(x+7) = 8\). First, expand the left-hand side to obtain a quadratic expression:\[ x^2 + 7x = 8 \]

Step 2: Rearrange the equation

Move all terms to one side of the equation to set it equal to zero.\[ x^2 + 7x - 8 = 0 \]

Step 3a: Factoring the quadratic equation

Look for two numbers that multiply to \(-8\) (the constant term) and add to \(7\) (the linear coefficient). The numbers are \(8\) and \(-1\).So, the factorization is:\[ (x + 8)(x - 1) = 0 \]

Step 4a: Solve using the factored form

Set each factor equal to zero and solve for \(x\):1. \(x + 8 = 0\) gives \(x = -8\)2. \(x - 1 = 0\) gives \(x = 1\)Thus, the solutions are \(x = -8\) and \(x = 1\).

Step 3b: Completing the square

Start with the equation:\[ x^2 + 7x - 8 = 0 \]Move the constant term to the other side to focus on completing the square:\[ x^2 + 7x = 8 \]

Step 4b: Find the constant to complete the square

Take half of the coefficient of \(x\), square it, and add to both sides. Half of \(7\) is \(\frac{7}{2}\), and its square is \(\frac{49}{4}\):\[ x^2 + 7x + \frac{49}{4} = 8 + \frac{49}{4} \]

Step 5b: Simplify the equation

The left-hand side becomes a perfect square:\((x + \frac{7}{2})^2 = \frac{81}{4}\)

Step 6b: Solve the equation

Take the square root of both sides:\[ x + \frac{7}{2} = \pm\frac{9}{2} \]This yields two equations:1. \(x + \frac{7}{2} = \frac{9}{2}\) leading to \(x = 1\)2. \(x + \frac{7}{2} = -\frac{9}{2}\) leading to \(x = -8\)Thus, the solutions are \(x = -8\) and \(x = 1\).

Key Concepts

Factoring Method

The factoring method is a great way to solve quadratic equations like the one given: \(x(x+7) = 8\). Start by expanding the left side of the equation: \(x^2 + 7x = 8\).
To apply the factoring method, we need to rearrange the equation to set it equal to zero. This involves moving all terms to one side: \(x^2 + 7x - 8 = 0\).
Once in this form, we look for two numbers that multiply to \(-8\) (the constant term) and add to \(7\) (the linear coefficient). These numbers are \(8\) and \(-1\), allowing us to write the equation as \((x+8)(x-1) = 0\).
To find the solutions, set each factor equal to zero:

• \(x + 8 = 0\) gives \(x = -8\)
• \(x - 1 = 0\) gives \(x = 1\)

Thus, the solutions are \(x = -8\) and \(x = 1\). Factoring can be quick and straightforward when the quadratic is easily decomposable into integer factors.

Completing the Square

Another effective way to solve quadratic equations is the method of completing the square. This involves converting the equation \(x^2 + 7x - 8 = 0\) into a perfect square trinomial.
Start by isolating the quadratic and linear terms on one side: \(x^2 + 7x = 8\). We will focus on these terms to form a perfect square trinomial.
To complete the square, take half of the coefficient of \(x\) (which is \(7\)), square it, and add it to both sides of the equation. Half of \(7\) is \(\frac{7}{2}\), and its square is \(\frac{49}{4}\).
Add \(\frac{49}{4}\) to both sides to get:\[x^2 + 7x + \frac{49}{4} = 8 + \frac{49}{4}\] The left side becomes \((x + \frac{7}{2})^2\), transforming our equation into: \[(x + \frac{7}{2})^2 = \frac{81}{4}\]
The next step is to solve for \(x\) by taking the square root of both sides, remembering the \(\pm\) sign:

• \(x + \frac{7}{2} = \frac{9}{2}\) leads to \(x = 1\)
• \(x + \frac{7}{2} = -\frac{9}{2}\) leads to \(x = -8\)

This method not only helps find solutions but also reveals the vertex form of the quadratic.

Quadratic Formula

The quadratic formula is a universal method suitable for solving any quadratic equation of the form \(ax^2 + bx + c = 0\). It is derived from completing the square, providing a direct route to the solutions. The formula is:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
To use the quadratic formula, identify the coefficients \(a\), \(b\), and \(c\) from the equation. For our problem \(x^2 + 7x - 8 = 0\), we have:

• \(a = 1\)
• \(b = 7\)
• \(c = -8\)

Substitute these values into the formula:\[x = \frac{-7 \pm \sqrt{(7)^2 - 4(1)(-8)}}{2(1)}\] Calculate the discriminant inside the square root, \(49 + 32 = 81\), and then solve:\[x = \frac{-7 \pm \sqrt{81}}{2}\] This gives two possible solutions:

• \(x = \frac{-7 + 9}{2} = 1\)
• \(x = \frac{-7 - 9}{2} = -8\)

The beauty of the quadratic formula lies in its applicability to all quadratic equations, ensuring that you can find solutions even when factoring is difficult or impossible.

Quadratic Equations

Quadratic equations are polynomial equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\).
These equations graph as parabolas, which can open upwards or downwards depending on the sign of \(a\).
Solving quadratic equations is central to algebra, and they can arise in various scenarios from physics to finance. The core methods for solving them include factoring, completing the square, and the quadratic formula, each having unique benefits.
- **Factoring** is often the quickest method when the equation can be easily decomposed.- **Completing the Square** helps with understanding vertex form and is perfect for precise solutions.- **Quadratic Formula** provides a surefire way to find solutions regardless of the equation's complexity. Understanding these concepts enhances mathematical problem-solving skills, providing techniques that apply beyond simple quadratic equations.