Problem 9 - Algebra 1: Concepts and Skills

Question

Tell whether the statement is true or false. If the statement is false, rewrite the right-hand side to make the statement true. $$ (3 x+4)^{2}=9 x^{2}+12 x+16 $$

Step-by-Step Solution

Answer

The given statement is false. The correct equation should be \((3x+4)^2=9x^2+24x+16\).

Step 1: Identifying the Equation

The given equation is \((3x+4)^2=9x^2+12x+16\). Now, this needs to be verified to see whether it is true or false.

Step 2: Expand the left-hand side of the Equation

Expand the left-hand side of the equation by using the formula \((a+b)^2 = a^2 + 2ab + b^2\). Applying the same rule, we have \((3x+4)^2=(3x)^2 + 2*3x*4 + 4^2 = 9x^2 +24x+16\).

Step 3: Compare both Sides and Check if the Equation is True or False

Upon expansion, the left side of the equation has turned out to be \(9x^2 + 24x + 16\). This is not equal to the right-hand side of the equation given, hence the statement is false.

Step 4: Rewrite the Equation

The equation should have been written as \((3x+4)^2=9x^2+24x+16\). This is the correct equation, and it is obtained after expanding the left-hand side of the given statement.

Key Concepts

Polynomial Expansion

When you expand a polynomial, you are breaking it down into a sum of terms. For example, expanding \((3x + 4)^2\) means multiplying the expression by itself: \((3x + 4) \times (3x + 4)\). This helps simplify and analyze the expression more easily.

• Use the rules of arithmetic to multiply each term.
• Combine like terms to simplify the expanded expression.

In this exercise, expanding \((3x + 4)^2\) involves applying specific algebraic properties.

Binomial Theorem

The binomial theorem offers a quick way to expand binomials raised to a power. It provides a formula that can be applied to expressions of the form \((a + b)^n\).

• The basic rule: \((a + b)^2 = a^2 + 2ab + b^2\).
• It helps simplify the process of expansion without needing to multiply manually.

When applied to \((3x + 4)^2\), you substitute \(a = 3x\) and \(b = 4\), resulting in \(9x^2 + 24x + 16\). This is key for verifying our equation.

Equation Verification

Verifying an equation is all about checking whether both sides are equal. This ensures the equation is true or if it needs to be corrected.

• Expand the expression as needed to observe all terms.
• Compare the expanded form with the original right-hand side of the equation.

In this case, after expanding \((3x+4)^2\), we found \(9x^2 + 24x + 16\), which was not equivalent to the given \(9x^2 + 12x + 16\), making the initial statement false. Revising the equation makes sure both sides match perfectly.