Problem 16 - Algebra 1: Concepts and Skills

Question

Solve the equation by factoring. $$ 7 x^{2}+28 x+28=0 $$

Step-by-Step Solution

Answer

The solution to the equation \(7x^{2}+28x+28=0\) by factoring is \(x = -2\).

Step 1: Rewrite the equation in standard form

Confirm that the given equation \( 7x^{2}+28x+28=0 \) is in a standard form which is \(ax^{2} + bx + c = 0 \)

Step 2: Factor out the Greatest Common Factor (GCF)

In this equation, the GCF is 7. Factor out 7 from all terms of the equation to simplify which leads to \( 7(x^{2}+4x+4)= 0\)

Step 3: Factor the quadratic

Factorize the quadratic \(x^2 + 4x + 4\). This equation factors to \( (x+2)^2\). This gives us \(7(x+2)^2 = 0\)

Step 4: Use the Zero product property

The Zero product property states that if the product of multiple factors equals zero, then at least one of the factors must be zero. Set the factor \(x+2\) equal to zero and solve for x which gives \(x = -2\)

Key Concepts

Greatest Common Factor

When solving quadratic equations by factoring, the first crucial step is often to find the Greatest Common Factor (GCF). The GCF is the largest number that can evenly divide all terms in an equation. Identifying and factoring out the GCF makes the equation simpler and easier to manage. To find the GCF, examine each term:

• Inspect the coefficients and constant terms to determine their shared divisor.
• In our exercise, the equation is \(7x^2 + 28x + 28 = 0\).
• The terms are: 7, 28, and 28. Notice they all can be divided by 7.

Factoring out 7 from each term, simplifies the equation to \(7(x^2 + 4x + 4) = 0\). This makes it much easier to see the underlying quadratic. Always simplify first by removing the GCF before continuing to solve the quadratic. This simplification can reveal important patterns and make subsequent steps considerably more straightforward.

Zero Product Property

Once you've factored out common terms and simplified the quadratic equation, the Zero Product Property comes into play. This property is an essential shortcut in mathematics, stating that if the product of multiple factors equals zero, then at least one factor must be zero. This rule allows us to set each factor containing the variable equal to zero.After factoring, the equation looks like this: \(7(x+2)^2 = 0\). Here, two factors are impacting the equation: the number 7 and the expression \((x+2)^2\). Based on the Zero Product Property:

• If \(A \cdot B = 0\), then \(A = 0\) or \(B = 0\), or both.
• Applying this, 7 is never 0, so we focus on \((x+2)^2 = 0\).
• Solve \((x+2)^2 = 0\) by setting \((x+2)\) to zero: \(x + 2 = 0\).
• This simplifies to find \(x = -2\).

The Zero Product Property is the reason why factoring is so powerful in solving quadratic equations.

Solving Quadratic Equations

Solving quadratic equations involves manipulating the equation into a form where finding solutions is straightforward. Factoring and using properties like the ones discussed simplifies even complex quadratics:The equation provided in the exercise \(7x^2 + 28x + 28 = 0\), though initially difficult, becomes manageable through systematic steps:

• First, confirm the equation is in standard form: \(ax^2 + bx + c = 0\).
• Identify the GCF, here it is 7, and factor it out to simplify the equation.
• Rewrite the equation: \(7(x^2 + 4x + 4) = 0\).
• Look at the simplified quadratic \(x^2 + 4x + 4\), which can directly factor into \((x + 2)^2\).
• Apply the Zero Product Property to solve: set \(x + 2 = 0\) which results in \(x = -2\).

Through these steps, we find the roots of the quadratic equation effectively. This approach can be generalized to solve any quadratic equation by using factoring when possible, supplemented by the Zero Product Property.