Question

When \((2 x-3)^{2}\) is written in the form \(a x^{2}+b x+c,\) where \(a, b,\) and \(c\) are integers, \(a+b+c=?\) A. \(-17\) B. \(-5\) C. 1 D. 13 E. 25

Short answer

Question: Determine the sum of the coefficients of the expanded form of the expression \((2x-3)^2\). Answer: 1

Step-by-step solution

Expand the expression

To expand the expression \((2x-3)^2\), first apply the formula for the square of a binomial: \((a-b)^2 = a^2 - 2ab + b^2\). So, we have: \((2x-3)^2 = (2x)^2 - 2(2x)(3) + (3)^2\) Now apply the multiplication laws to get the expanded form: \(4x^2 - 12x + 9\)

Identify the coefficients

We can see that the given expression expanded is in the form \(ax^2 + bx + c\). Therefore, we can identify the coefficients accordingly: \(a = 4, b = -12, c = 9\)

Calculate the sum of the coefficients

Now that we have the coefficients, let's add them together to find \(a+b+c\): \(a + b + c = 4 + (-12) + 9 = -8 + 9 = 1\) Hence, \(a+b+c=1\), which corresponds to answer choice C.

Key concepts

Expanding Binomials

Expanding binomials is essential in simplifying algebraic expressions and solving equations. A binomial is an algebraic expression containing two terms. For instance, in the exercise \( (2x-3)^2 \), the binomial is \(2x-3\), and we are interested in finding its square. Expanding a binomial like this involves applying the formula \( (a-b)^2 = a^2 - 2ab + b^2 \) which is derived from the distributive property of multiplication over addition and subtraction.

Let's break it down: The square of the first term, \( (2x)^2 \) gives us \( 4x^2 \). The product of the terms, multiplied by 2, \( -2(2x)(3) \) results in \( -12x \). Finally, the square of the last term, \( (3)^2 \) is \( 9 \). Combining these, the expanded form of the binomial \( (2x-3)^2 \) becomes \( 4x^2 - 12x + 9 \). This process transforms a compact binomial expression into a format that is often easier to work with in equations.

Quadratic Expressions

Quadratic expressions are polynomials of degree two, typically written in the standard form \( ax^2 + bx + c \) where \( a \), \( b \) and \( c \) are constants. In our example, once the binomial \( (2x-3)^2 \) is expanded, the result is the quadratic expression \( 4x^2 - 12x + 9 \).

These expressions are fundamental when dealing with parabolic graphs, finding the axis of symmetry, and solving for the roots or zeroes of a function. Each part of the quadratic expression holds significance: \( a \) determines the opening direction and the width of the parabola, \( b \) affects the position of the vertex along the x-axis, and \( c \) represents the y-intercept of the graph. Understanding the structure of these expressions enables students to analyze and graph quadratic functions effectively.

Coefficient Calculations

Coefficient calculations involve finding the numerical factors that multiply the variables in an algebraic expression. In the context of expanding binomials and quadratic equations, identifying coefficients is crucial to understanding the expression's behavior. After expanding \( (2x-3)^2 \) to \( 4x^2 - 12x + 9 \), we can see the coefficients clearly: \( a = 4 \), \( b = -12 \), and \( c = 9 \).

In many algebraic procedures, such as solving, graphing, or simplifying expressions, knowing these coefficients is the first step. Following identification, one can perform further calculations, like finding the sum of coefficients as in our exercise. Adding these numerical values, despite being straightforward, plays a central role when working with polynomials. Therefore, the sum \( a + b + c = 4 - 12 + 9 \) equals \( 1 \) for the given quadratic, proving that meticulous coefficient calculations are indispensable for accurate algebraic manipulation.