Problem 45 - Algebra and Trigonometry with Analytic Geometry

Question

Show that the equation is an identity. $$(4 x-3)^{2}-16 x^{2}=9-24 x$$

Step-by-Step Solution

Answer

Yes, the equation is an identity: both sides are equal.

Step 1: Expand the Expression on the Left

First, expand the term \((4x - 3)^2\). Recall that \((a - b)^2 = a^2 - 2ab + b^2\). Here, \(a = 4x\) and \(b = 3\), giving: \[(4x - 3)^2 = (4x)^2 - 2(4x)(3) + 3^2 = 16x^2 - 24x + 9\] So, the equation becomes: \[16x^2 - 24x + 9 - 16x^2 = 9 - 24x\]

Step 2: Simplify Left Side by Combining Like Terms

Now, simplify the left side by combining like terms: Since \(16x^2 - 16x^2 = 0\), the left side becomes: \[-24x + 9\] Thus, the equation now is: \[-24x + 9 = 9 - 24x\]

Step 3: Compare Both Sides of the Equation

Observe and compare both sides of the simplified equation: Left Side: \[-24x + 9\] Right Side: \[9 - 24x\] Since the left side matches the right side, \(-24x + 9 = 9 - 24x\), the equation is an identity.

Key Concepts

Expanding Expressions

In the realm of algebra, expanding expressions involves removing brackets from terms that are multiplied together, following specific rules or formulas. In our exercise, we are expanding the term \((4x - 3)^2\). This follows the algebraic identity for the square of a binomial: \((a - b)^2 = a^2 - 2ab + b^2\). Here, \(a = 4x\) and \(b = 3\).
To start expanding, we square \(a\) to get \((4x)^2 = 16x^2\). Next, we multiply \(a\) and \(b\) together: \(2(4x)(3) = 24x\), and don't forget the negative sign, as in the formula it's \(-2ab\), giving us \(-24x\). Finally, square \(b\) to get \(3^2 = 9\). All these yield the expanded expression:

• \((4x)^2 - 2(4x)(3) + 3^2 = 16x^2 - 24x + 9\).

This expansion helps open up the equation, making further steps like simplification easier to manage.

Simplification

Simplification in algebra refers to the process of making an expression or equation easier to work with. In our exercise, after expanding \((4x - 3)^2\), we arrived at \(16x^2 - 24x + 9 - 16x^2\). Simplification involves combining like terms by performing basic arithmetic operations.
To simplify, look at the terms on the left side of the equation. Notice that \(16x^2 - 16x^2 = 0\) because they cancel each other out. You’re left with:

• \(-24x + 9\).

By combining these like terms, you effectively simplify the expression which matches the right side \(9 - 24x\). This process is essential as it reveals whether both sides of the equation are identical, confirming the identity.

Algebraic Identities

An algebraic identity is an equation that holds true for all values of the involved variables. In this context, proving that the equation \((4x-3)^{2}-16x^{2}=9-24x\) is an identity means showing both sides of the equation are consistently equal.
By first expanding and then simplifying the left side of the equation, you end with \(-24x + 9\). Observe that this is exactly identical to the right side \(9 - 24x\). By this observation, we verify the identity because every algebraic step confirms equality:

• Left side: \(-24x + 9\).
• Right side: \(9 - 24x\).

The process of working through an equation to verify it's an identity reinforces understanding of algebraic principles, illustrating how steps like expansion and simplification are interlinked for solving such problems.