Problem 8 - Algebra for College Students

Question

Solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. $$x^{2}+7 x-30=0$$

Step-by-Step Solution

Answer

The solutions are \(x = -10\) and \(x = 3\).

Step 1: Identify the equation structure

The given equation is a quadratic equation of the form \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = 7 \), and \( c = -30 \). This indicates that it is a trinomial that could potentially be factored into a product of two binomials.

Step 2: Determine the factors of the constant term

Since the constant term \( c = -30 \) is negative, we need to find two numbers that multiply to \(-30\) and add up to \(7\) (the coefficient of the middle term). We list pairs of factors of \(-30\) and identify \(10\) and \(-3\) as the numbers since \(10 imes -3 = -30\) and \(10 + (-3) = 7\).

Step 3: Write the equation in factored form

Using the numbers identified in Step 2, rewrite the quadratic expression \( x^2 + 7x - 30 \) in its factored form: \( (x + 10)(x - 3) = 0 \).

Step 4: Solve for the values of x

Apply the zero product property: if \((A)(B) = 0\), then \(A = 0\) or \(B = 0\). Set each factor obtained in Step 3 equal to zero: \(x + 10 = 0\) and \(x - 3 = 0\).

Step 5: Calculate solutions from factored form

Solve the equations from Step 4: For \(x + 10 = 0\), subtract 10 from both sides to get \(x = -10\). For \(x - 3 = 0\), add 3 to both sides to get \(x = 3\). These are the solutions of the original equation.

Key Concepts

Factoring Techniques

Factoring techniques are essential tools when working with quadratic equations. The primary goal is to express the original quadratic equation in a simpler form, generally a product of two binomials. This makes it easier to find the roots or solutions of the equation.
To factor a quadratic expression like \(x^2 + 7x - 30\), follow these steps:

• Identify the coefficients \(a\), \(b\), and \(c\) from the standard form \(ax^2 + bx + c\).
• The focus is on finding two numbers that multiply to the constant \(c\) (here, \(-30\)) and add to \(b\) (here, \(7\)).
• List potential pairs of factors of \(-30\). Each pair includes one positive and one negative number because their product must be negative.
• Amongst these, pinpoint the pair \(10\) and \(-3\) as they fit both conditions (\(10 \times -3 = -30\) and \(10 + (-3) = 7\)).

With the correct pair of factors, rewrite the quadratic as a product of binomials: \((x + 10)(x - 3)\). This structure is much easier to work with for solving the equation.

Zero Product Property

The zero product property is a fundamental principle in solving quadratics. It states that if the product of two factors equals zero, then at least one of the factors must be zero.
This property is used once a quadratic equation is factored. Given \((x + 10)(x - 3) = 0\), you can apply the zero product property by setting each factor to zero individually:

• \(x + 10 = 0\)
• \(x - 3 = 0\)

By solving these separate equations, you find the potential values of \(x\) that make the original quadratic equation true. This method is powerful because it breaks the quadratic problem into simpler, linear equations, which are generally easier to solve.

Solving Quadratics

Solving quadratic equations involves finding all the values of \(x\) that satisfy the equation. Using the combination of factoring techniques and the zero product property is one effective strategy.
Once the quadratic is factored into \((x + 10)(x - 3) = 0\), solving it involves individual consideration of each binomial:

• For \(x + 10 = 0\), subtract 10 from both sides to isolate \(x\) and find \(x = -10\).
• For \(x - 3 = 0\), add 3 to both sides to isolate \(x\) and find \(x = 3\).

These operations are straightforward, turning the task of solving a quadratic into a process involving manageable steps.
Therefore, the solutions to the equation \(x^2 + 7x - 30 = 0\) are \(x = -10\) and \(x = 3\). Recognizing patterns and following structured steps allows for efficient and accurate solutions in quadratic problems.