Problem 9 - Algebra 2

Question

Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square. \(x^{2}-3 x+c\)

Step-by-Step Solution

Answer

The value of c is \(\frac{9}{4}\), and the trinomial is \(\left(x - \frac{3}{2}\right)^2\).

Step 1: Understanding Perfect Square Trinomials

A perfect square trinomial is a quadratic expression of the form \(a^2 + 2ab + b^2 = (a + b)^2\). Therefore, we can see that \(x^2 - 3x + c\) needs to match this pattern, where \(a = x\) and \(2ab = -3x\).

Step 2: Identify the Middle Term

The given expression is \(x^2 - 3x+c\). Here, the middle term is -3x, which corresponds to \(2ab\) in the perfect square trinomial format, where \(a = x\). This means \(-3 = 2b\).

Step 3: Solve for b

To find \(b\), solve \(-3 = 2b\): \(b = -\frac{3}{2}\).

Step 4: Find c using b

Since \(c = b^2\) in a perfect square trinomial, substitute the value of \(b\):\(c = \left(-\frac{3}{2}\right)^2 = \frac{9}{4}\).

Step 5: Write as a Perfect Square

Now that we have \(c = \frac{9}{4}\), the expression \(x^2 - 3x + \frac{9}{4}\) can be written as a squared binomial:\(\left(x - \frac{3}{2}\right)^2\).

Key Concepts

Quadratic Expressions

In algebra, a quadratic expression is a polynomial expression of degree two, commonly written in the form \( ax^2 + bx + c \). This form features three terms:

• The quadratic term \( ax^2 \)
• The linear term \( bx \)
• The constant term \( c \)

Quadratic expressions are fundamental in mathematics because they can describe a variety of physical phenomena. For our exercise, the expression \( x^2 - 3x + c \) is a quadratic expression. Here, the leading coefficient \( a \) is 1, the middle term coefficient \( b \) is -3, and \( c \) is the term we're solving for.
Quadratic expressions can sometimes be factored or rewritten as perfect square trinomials. A perfect square trinomial takes the form of a square of a binomial, according to the equation \( a^2 + 2ab + b^2 = (a + b)^2 \). Knowing how to manipulate quadratics into this form is a valuable skill in algebra.

Solving Equations

Solving equations involves finding the values of unknowns that make the equation true. In our task, we aim to find the value of \( c \) that transforms the quadratic expression \( x^2 - 3x + c \) into a perfect square trinomial.
First, identify the requirement to fulfill: the expression must match the pattern \( a^2 + 2ab + b^2 \). The challenge is to match the given expression with the perfect square form. Since our expression is \( x^2 - 3x + c \), it equates to \( a^2 \) plus a middle term, leaving us to solve for the unknown \( c \).

• Recognize the middle term \( -3x \) that corresponds to \( 2ab \).
• From \( 2ab = -3x \), solve for \( b \) using \( -3 = 2b \), giving \( b = -\frac{3}{2} \).

Finally, compute \( c = b^2 \).
By solving this equation, you find \( c = \left(-\frac{3}{2}\right)^2 = \frac{9}{4} \). Thus, \( x^2 - 3x + \frac{9}{4} \) becomes \( \left( x - \frac{3}{2} \right)^2 \), confirming it as a perfect square trinomial.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying algebraic expressions to make them easier to solve or to evaluate. In this exercise, algebraic manipulation helps us transform \( x^2 - 3x + c \) into a recognizable perfect square trinomial form.

• First, understand the needed structure: \( a^2 + 2ab + b^2 \).
• Identify components by matching coefficients. Recognize \( 2ab = -3x \), allowing us to solve for \( b \).
• Once \( b \) is determined as \(-\frac{3}{2} \), leverage prior knowledge that \( c = b^2 \).

After performing these steps, you use algebraic manipulation to transform the trinomial into its perfect square form: \( \left( x - \frac{3}{2} \right)^2 \).
Mastering algebraic manipulation enables solving complex problems, helps in problem-solving strategies, and forms the cornerstone of higher mathematics, paving the way for calculus and beyond.