Problem 56 - College Algebra Essentials

Question

By making an appropriate substitution. $$(x+3)^{2}+7(x+3)-18=0$$

Step-by-Step Solution

Answer

The solutions to the original equation are \( x = -1, -12 \).

Step 1: Perform a substitution

As per the outline, substitute \( u \) in place of \( x + 3 \) in the given equation. So, it becomes \( u^{2} + 7u - 18 = 0 \).

Step 2: Solve the quadratic equation

Now, solve this quadratic equation for \( u \). By factoring \( u^2 + 7u - 18 = 0 \), we obtain \( (u-2)(u+9) = 0 \). This gives roots \( u = 2, -9 \).

Step 3: Substitute back to find x

Now, substitute \( u \) back into \( x + 3 \). So, \( x + 3 = 2 \) which yields \( x = -1 \) and \( x + 3 = -9 \) which yields \( x = -12 \).

Key Concepts

Understanding Quadratic Equations

A quadratic equation is a fundamental concept in algebra, describing a polynomial equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The highest exponent of the variable \( x \) is 2, which makes it a true quadratic.Quadratic equations can be identified by their characteristic "U" shaped curve called a parabola, which can open upwards or downwards depending on the sign of \( a \). Solving a quadratic equation can give you two solutions, which are also known as roots or zeroes of the equation.Key methods to solve quadratic equations include:

• Factoring: Expressing the equation as a product of its linear factors.
• Quadratic Formula: Using the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the roots.
• Completing the Square: Rewriting the quadratic equation in a perfect square form.
• Graphically: Identifying the points where the parabola intersects the x-axis.

Understanding these methods helps deepen comprehension and improves problem-solving skills in algebra.

The Art of Factoring Quadratics

Factoring is a method used to solve quadratic equations by expressing them as a product of two or more polynomials. This method is efficient and provides the roots of the quadratic equation with ease. Consider the quadratic equation \( u^2 + 7u - 18 = 0 \) from our example. Here's how you factor a quadratic:

• Start by identifying two numbers that multiply to the constant term (\(-18\)) and add to the coefficient of the linear term (\(7\)).
• For this equation, the numbers \(2\) and \(-9\) work because \(2 \times -9 = -18\) and \(2 + 9 = 7\).
• Rewrite the equation as \((u - 2)(u + 9) = 0\).
• Solve for \(u\) by setting each factor equal to zero: \(u - 2 = 0\) or \(u + 9 = 0\), giving the roots \(u = 2\) and \(u = -9\).

Factoring is widely used because it simplifies algebraic expressions and provides clear solutions. It strengthens algebraic manipulation skills and reinforces understanding of polynomial structure.

Crafting an Algebraic Solution

The substitution method is a powerful tool in algebra for solving complex equations by reducing them into simpler forms. In the exercise, by substituting \( u = x + 3 \), the original complex equation \((x+3)^{2}+7(x+3)-18=0\) is transformed into a simpler one, \( u^2 + 7u - 18 = 0 \). This makes the process more intuitive and manageable.Steps for using the substitution method effectively:

• Identify a part of the equation that can be simplified by substitution. Choose a new variable to replace it, such as \( u = x + 3 \).
• Substitute this new variable into the equation to reduce its complexity.
• Solve the resulting equation using appropriate methods, like factoring in this example.
• Finally, substitute back to resolve the original variables, as \( u = x + 3 \) gives the solutions for \( x \).

Understanding the algebraic solution using substitution can transform difficult problems into straightforward tasks. It highlights the importance of strategic thinking and enhances logical reasoning abilities in algebra.