Problem 7 - Intermediate Algebra

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

First, we expand the given quadratic equation. Start by distributing the variable \(x\) on the left-hand side: \[x(x + 7) = 8\] becomes \[x^2 + 7x = 8\].

Step 2: Rearrange the Equation

Bring all terms to one side to set the equation to zero: \[x^2 + 7x - 8 = 0\]. This is the standard form of a quadratic equation \(ax^2 + bx + c = 0\).

Step 3: Factoring Method - Identify Factors

To factor the quadratic equation \(x^2 + 7x - 8 = 0\), look for two numbers that multiply to \(-8\) and add to \(7\). These numbers are \(8\) and \(-1\).

Step 4: Factoring Method - Write the Factored Equation

Based on the identified factors, express the equation as: \[(x + 8)(x - 1) = 0\].

Step 5: Solve Using Factoring

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

Step 6: Completing the Square - Start with the Original Form

Using \(x^2 + 7x - 8 = 0\), add \(8\) to both sides to isolate the quadratic and linear terms: \[x^2 + 7x = 8\].

Step 7: Completing the Square - Find the Perfect Square

Take half of the coefficient of \(x\) (which is \(7/2\)), square it, and add it to both sides:\[(7/2)^2 = 49/4\]Thus, add \(49/4\) to both sides: \[x^2 + 7x + 49/4 = 8 + 49/4\].

Step 8: Simplify Completing the Square

The left side becomes a perfect square: \[(x + 7/2)^2\]. On the right side, convert \(8\) to \(32/4\) and add: \[8 + 49/4 = 32/4 + 49/4 = 81/4\]. So, the equation is now \[(x + 7/2)^2 = 81/4\].

Step 9: Solve Using Completing the Square

Take the square root of both sides:\[x + 7/2 = \pm \sqrt{81/4}\], which simplifies to \(x + 7/2 = \pm 9/2\). Thus, solve for \(x\):1. \(x + 7/2 = 9/2\) yields \(x = 1\).2. \(x + 7/2 = -9/2\) yields \(x = -8\).The solutions are \(x = -8\) and \(x = 1\).

Key Concepts

Factoring Method

Quadratic equations can often be solved using the factoring method. This is popular because it offers a straightforward path to finding solutions. Here's how it works for the equation \(x^2 + 7x - 8 = 0\).
The first step is to express the equation in standard form, as given in the problem statement. Next, we look for two numbers that multiply to the constant term (-8) and add to the linear coefficient (7). These two numbers are 8 and -1.

• Numbers found: 8 and -1
• Multiply: 8 \( \times \) -1 = -8
• Add: 8 + (-1) = 7

Now, substitute these numbers into their respective factors: \((x + 8)(x - 1) = 0\). Using the zero product property, set each factor equal to zero:

• \(x + 8 = 0\) leads to \(x = -8\)
• \(x - 1 = 0\) leads to \(x = 1\)

Thus, the solutions we obtain are \(x = -8\) and \(x = 1\). These are the roots of the equation determined via factoring.

Completing the Square

Completing the square is another effective method to solve quadratic equations. It involves forming a perfect square trinomial to make the equation easier to solve. Starting with our equation \(x^2 + 7x - 8 = 0\):
First, isolate the quadratic and linear terms by adding 8 to both sides: \(x^2 + 7x = 8\). Next, take half of the linear coefficient (7), square it, and add it to both sides of the equation:

• Half of 7 is \(\frac{7}{2}\)
• Square it: \(\left(\frac{7}{2}\right)^2 = \frac{49}{4}\)

Add \(\frac{49}{4}\) to both sides, so the left side becomes a perfect square: \(x^2 + 7x + \frac{49}{4} = 8 + \frac{49}{4}\). Simplifying the right side, we convert 8 to \(\frac{32}{4}\) and add:
\[x^2 + 7x + \frac{49}{4} = \frac{81}{4}\]
This left side can be rewritten as \((x + \frac{7}{2})^2\), leading to the equation \((x + \frac{7}{2})^2 = \frac{81}{4}\). Now, solve by taking the square root of both sides:

• \(x + \frac{7}{2} = \pm \frac{9}{2}\)

Solving gives the roots:

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

Thus, we again find solutions \(x = -8\) and \(x = 1\).

Roots of Polynomial

In mathematics, the roots of a polynomial are the values of the variable that satisfy the given polynomial equation. In simpler terms, these are the solutions which make the polynomial equation equal to zero. For a quadratic equation like \(x^2 + 7x - 8 = 0\), finding the roots involves a few interesting methods like factoring and completing the square, as discussed earlier.
Roots tell us important points of intersection of a polynomial with the x-axis on a graph. Understanding the roots helps us predict the behavior of the quadratic function in real-world applications, such as determining times when a projectile will hit the ground based on the height equation (a quadratic).

• Roots are technically the 'zeroes' of the function.
• They are the x-values where the graph touches or crosses the x-axis.
• For quadratic equations, there can be 0, 1, or 2 real roots.

In our solved equation \(x^2 + 7x - 8 = 0\), the roots are \(x = -8\) and \(x = 1\). These solutions indicate the points where the polynomial will cross the x-axis.