Problem 3 - Beginning and Intermediate Algebra

Question

Based on the discussion and examples of this section, give the first step to solve each equation. Do not actually solve. \(\left(x^{2}+x\right)^{2}-8\left(x^{2}+x\right)+12=0\)

Step-by-Step Solution

Answer

Let y = \(x^2 + x\) and substitute it in the equation.

Step 1: Identify the substitution variable

Recognize the repetitive expression in the equation. Let \ y = \(x^2 + x\). This substitution simplifies the equation significantly.

Step 2: Substitute the variable

Replace \(x^2 + x\) with y. The equation becomes \ y^2 - 8y + 12 = 0.

Key Concepts

quadratic equations

Quadratic equations are polynomials of degree 2. This means their highest exponent is 2. A typical form of a quadratic equation is ax^2 + bx + c = 0 where:

• a, b, and c are constants
• x represents the variable

In our case, after the substitution, we get y^2 - 8y + 12 = 0. This fits the quadratic equation form, where:

• a = 1
• b = -8
• c = 12

Solving quadratic equations typically involves:

• Factoring
• Using the quadratic formula
• Completing the square

Each method has its purpose and is chosen based on the equation.

repetitive expressions

Repetitive expressions in algebra are parts of equations that repeat and can be simplified. For example, in our original equation: (x^2 + x)^2 - 8(x^2 + x) + 12 = 0
The expression x^2 + x repeats. This repetition can make solving the equation tedious if approached directly. Recognizing these can help simplify and solve equations efficiently.

By substituting these repetitive expressions with a single variable, we reduce complexity. This is a crucial strategy in algebra, especially with higher-degree polynomials.
Doing so transforms an initially challenging problem into a more manageable form.

substitution variable

A substitution variable is a placeholder for a repetitive expression. In our equation, we let y = x^2 + x.
This substitution helps simplify the equation from: (x^2 + x)^2 - 8(x^2 + x) + 12 = 0 to y^2 - 8y + 12 = 0
This step breaks down a complicated expression into something easier to handle.

Substitution variables are generally chosen for expressions that:

• Appear multiple times within an equation
• Can be isolated easily

Once the equation with the substitution variable is solved, the final step is to substitute back to find the solution in terms of the original variable.