Problem 40 - College Algebra Essentials

Question

In Exercises \(35-46,\) determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then write and factor the trinomial. $$ x^{2}+5 x $$

Step-by-Step Solution

Answer

The constant that should be added to the binomial \(x^2+5x\) to make it a perfect square trinomial is 6.25. The resulting perfect square trinomial is \((x+5/2)^2\).

Step 1: Identify the terms

From the binomial \(x^2+5x\), we relate it with \(a^2+2ab\), thus \(a=x\), \(2ab=5x\). We can now solve for \(b\).

Step 2: Solve for b

The equation we get from above is \(2xb=5x\), if we divide both sides by \(2x\), assuming \(x!=0\), we can solve for \(b\). Therefore, \(b=5/2\).

Step 3: Find the Constant to be Added

Since \(b=5/2\), and we are looking for \(b^2\), we square b, \(b^2=(5/2)^2=6.25\).

Step 4: Convert to a perfect square trinomial

Now we add this constant to our original expression, so the trinomial becomes \(x^2+5x+6.25=(x+b)^2=(x+5/2)^2\)which is now a perfect square trinomial.

Step 5: Factorise the Trinomial

We represent the trinomial in factor form, this would be \((x+5/2)^2\). Thus, the binomial \(x^2+5x\) transforms into the perfect-square trinomial \((x+5/2)^2\) when we add the constant 6.25.

Key Concepts

Factoring Trinomials

Factoring trinomials involves breaking down quadratic expressions into simpler components. These components are typically products of two binomials, which, when multiplied out, result in the original trinomial. Understanding the process helps you recognize patterns and use them to solve more complex equations easily.

A trinomial is usually presented in the standard form of a quadratic equation, like so:

• \[ ax^2 + bx + c \]

To factor this trinomial, you look for two numbers that multiply to \(ac\) (the product of the leading coefficient and the constant term) and add up to \(b\) (the coefficient of the linear term).

In the example \(x^2 + 5x\), we add 6.25 to make it a perfect square trinomial, allowing it to factor into

• \((x + \frac{5}{2})^2\)

This demonstrates how additional terms help in factoring by making complex expressions more manageable.

Quadratic Expressions

Quadratic expressions form the backbone of many algebraic equations, characterized by the square of an unknown variable. The general form of a quadratic expression is

• \[ ax^2 + bx + c \]

where \(a\), \(b\), and \(c\) are constants. These expressions appear in various real-world applications, such as physics problems involving area or motion.

A common task involves transforming quadratic expressions to reveal insightful properties about their structure, often done through factoring or completing the square.

In our specific exercise, the quadratic "lead" was transformed into a perfect square trinomial from its original binomial form by determining a specific constant to add. Recognizing terms that complete these expressions is essential to simplification and solving.

Completing the Square

Completing the square is a technique used to convert quadratic expressions into a perfect square form. This transformation provides a clear view of the quadratic's properties, such as its vertex in graphing contexts, making it a valuable tool for solving equations.

The process involves determining what constant needs to be added to a binomial to make it a perfect square trinomial. In practice, this method is done by taking the coefficient of the linear term (\(b\)), halving it, and then squaring the result.

For example, with \(x^2 + 5x\), identify \(b\) as 5. Halve it to get \(\frac{5}{2}\), then square it to obtain \(\frac{25}{4} = 6.25\), which is the constant to add. This makes the expression become

• \(x^2 + 5x + 6.25 = (x + \frac{5}{2})^2\)

The resulting perfect square makes further algebraic manipulations straightforward, often simplifying solutions to quadratic equations significantly.