Problem 56 - College Algebra

Question

Solve each equation by making an appropriate substitution. $$(x+3)^{2}+7(x+3)-18-0$$

Step-by-Step Solution

Answer

The solutions to the given equation are \(x_{1}=-1\) and \(x_{2}=-12\).

Step 1: Apply substitution

Perform the substitution \(y=x+3\), which transforms the given equation into a simpler quadratic equation \((y)^{2}+7y-18=0\). This equation is easier to work with.

Step 2: Solve the quadratic equation

The quadratic equation \(y^{2}+7y-18=0\) can be solved using methods such as factoring, completing the square, or the quadratic formula. By factoring the equation we get: \((y-2)(y+9)=0\). Setting each factor equal to zero gives the solutions:\(y_{1}=2\), and \(y_{2}=-9\).

Step 3: Re-substitute to find x

Now substitute \(y=x+3\) back in to find the final solutions for x. Hence, the solutions to the equation are \(x_{1}=2-3=-1\) and \(x_{2}=-9-3=-12\).

Key Concepts

The Substitution Method to Simplify Quadratics

One of the critical strategies for solving quadratic equations is the substitution method. This approach is particularly beneficial when you're dealing with a complex quadratic equation that can be simplified into a more manageable form. In the exercise provided, the substitution method starts by identifying a substitution that will reduce the equation to a basic quadratic form. For the equation \(x+3)^2+7(x+3)-18=0\), we set \(y = x + 3\). This substitution creates a new equation, \(y^2 + 7y - 18 = 0\), which is clearly a quadratic equation and hence easier to solve.

The substitution method not only simplifies the equation but also makes it easier to apply other algebraic methods such as factoring or using the quadratic formula. It's important to remember, as shown in the step-by-step solution, that after solving for \(y\), you have to reverse the substitution to find the actual values of \(x\). This involves replacing \(y\) back with \(x + 3\) and solving the resulting equations to find the final solutions for \(x\).

Factoring Quadratics

Factoring quadratics is a fundamental technique for finding the roots of a quadratic equation without resorting to the quadratic formula. It is based on the principle that a quadratic equation can be expressed as the product of two binomial expressions. For our equation \(y^2 + 7y - 18 = 0\), factoring involves finding two numbers that multiply to give the constant term, \(-18\), and add up to the linear coefficient, \(7\).

Steps to Factor a Quadratic Equation

In this case, the numbers 9 and -2 fulfill this requirement since \(9 \times -2 = -18\) and \(9 + (-2) = 7\). Thus, the equation can be rewritten as \(y - 2)(y + 9) = 0\). The Zero Product Property then tells us that if a product of two factors is zero, at least one of the factors must be zero. This gives us two possible solutions: \(y - 2 = 0\) or \(y + 9 = 0\), leading to \(y_1 = 2\) and \(y_2 = -9\). Factoring is a powerful method that, when applicable, provides a straightforward path to solving quadratics.

The Quadratic Formula

When factoring is complex or impossible, the quadratic formula comes to the rescue. This formula provides a means to solve any quadratic equation, regardless of its form. The quadratic formula states that for a quadratic equation of the form \(ax^2 + bx + c = 0\), the solutions can be found using \(x = \frac{{-b \: \: \: \: \: \pm \: \: \: \: \: \: \sqrt{b^2-4ac}}}{{2a}}\). It is a foolproof method that uses the coefficients of the equation (\(a\), \(b\), and \(c\)) to find the solutions directly.

Even though our exercise can be solved by simpler methods like factoring, it's vital to understand how the quadratic formula works as a universal tool. It especially shines when the equation lacks easy-to-find factors or when dealing with imaginary numbers. Understanding and applying the quadratic formula ensures you will always have a method available for solving quadratic equations, making it an indispensable part of your mathematical toolkit.