Problem 48 - Algebra 2

Question

Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square. \(x^{2}-\frac{8}{3} x+c\)

Step-by-Step Solution

Answer

The value of c is \(\frac{16}{9}\), and the trinomial is \((x - \frac{4}{3})^2\).

Step 1: Understand a Perfect Square Trinomial

A perfect square trinomial is of the form \[ (x + a)^2 = x^2 + 2ax + a^2 \].This means that the trinomial \(x^2 - \frac{8}{3}x + c\) can be written as a perfect square if it matches this pattern.

Step 2: Identify 'a' from the Middle Term

Comparing \(x^2 - \frac{8}{3}x + c\) with \(x^2 + 2ax + a^2\), we can see that the middle term \(-\frac{8}{3}x\) corresponds to \(2ax\). This allows us to solve for 'a'. Set:\[-8/3 = 2a\]Divide both sides by 2:\[a = -\frac{8}{6} = -\frac{4}{3}\].

Step 3: Calculate the Value of c

Once we know \(a = -\frac{4}{3}\), we can find \(c\) because \(c = a^2\). Square \(-\frac{4}{3}\):\[a^2 = \left(-\frac{4}{3}\right)^2 = \frac{16}{9}\]Thus, \(c = \frac{16}{9}\).

Step 4: Write the Trinomial as a Perfect Square

Now that we have \(c = \frac{16}{9}\), our trinomial becomes:\[x^2 - \frac{8}{3}x + \frac{16}{9}\].This can be written as:\[(x - \frac{4}{3})^2\], since \((x - a)^2 = x^2 - 2ax + a^2\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators such as addition, subtraction, multiplication, and division. They form the language of algebra and are used to express mathematical relationships.

- A simple algebraic expression like \(x^2 - \frac{8}{3}x + c\) is built from: - The variable \(x\), which can represent different numbers. - Constants (like the given \(-\frac{8}{3}\) and \(c\)), which are fixed values.
Algebraic expressions can represent a wide variety of situations, from simple arithmetic calculations to complex geometric problems.

Expressing a problem as an algebraic expression allows us to use algebra's powerful tools and methods to solve it effectively. Recognizing patterns, like those found in perfect square trinomials, is a key skill in manipulating these expressions.

Completing the Square

Completing the square is a valuable technique used to solve quadratic equations and to rewrite expressions as perfect squares. It transforms a trinomial into a perfect square trinomial, making certain mathematical challenges more approachable.

Here's the main idea:

• Identify the coefficient of the middle term, \(- \frac{8}{3}x\). Here, it corresponds to \(2ax\) from the standard form \((x + a)^2 = x^2 + 2ax + a^2\).
• By comparing, set \(- \frac{8}{3} = 2a\) and solve for \(a\), which gives \(a = - \frac{4}{3}\).
• Next, calculate \(c\) using \(c = a^2\), resulting in \(c = \frac{16}{9}\).

Completing the square helps to simplify expressions and solve equations, especially when the equations do not factor easily. It is particularly useful in solving quadratic equations by converting them into a format that reveals the roots directly.

Quadratic Equations

Quadratic equations are key concepts in algebra, appearing frequently in both pure and applied mathematics. A general quadratic equation has the format \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(x\) is the variable. Our given trinomial \(x^2 - \frac{8}{3}x + c\) fits this form when \(c\) is specific, as derived in the exercise.

Quadratic equations can be solved in several ways:

• Factoring, when the equation can be expressed as the product of two binomials.
• Using the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), which gives solutions directly.
• Completing the square, which involves forming a perfect square trinomial.

Each method has its advantages, and completing the square can be used whether the quadratic is factorable or not, providing a versatile approach. Understanding these techniques is essential to mastering quadratic equations and tackling problems involving them successfully.