Problem 57 - Elementary Algebra

Question

The area of a square is given by the function \(A(x)=x 2-14 x+49,\) where \(x\) is measured in meters. Rewrite this function in factored form.

Step-by-Step Solution

Answer

The factored form is \((x - 7)^2\).

Step 1: Identify the Quadratic Expression

We start by identifying the quadratic expression given in the area function: \[ A(x) = x^2 - 14x + 49. \]We need to rewrite this expression in a factored form.

Step 2: Recognize the Perfect Square

Observe that the quadratic expression \(x^2 - 14x + 49\) resembles the expansion of a perfect square. In a perfect square, we have an expression of the form \((x - a)^2 = x^2 - 2ax + a^2\).

Step 3: Match the Quadratic Terms

Compare the expression \(x^2 - 14x + 49\) with \(x^2 - 2ax + a^2\). Here, \(-2a = -14\) and \(a^2 = 49\).

Step 4: Solve for \(a\)

From \(-2a = -14\), solve for \(a\):\[2a = 14 \a = \frac{14}{2} = 7.\]

Step 5: Verify Perfect Square Conditions

Now verify that \(a^2 = 49\) confirms \(a = 7\):\[a^2 = 7^2 = 49.\]Therefore, the expression \(x^2 - 14x + 49\) can be rewritten.

Step 6: Write in Factored Form

Since the expression satisfies the conditions for a perfect square, write it as:\[A(x) = (x - 7)^2.\]This is the factored form of the given quadratic expression.

Key Concepts

Perfect Square Trinomials

Perfect square trinomials are special types of quadratic expressions that can be rewritten as the square of a binomial. These are formed when a binomial is multiplied by itself. For example, if you have the expression \((x - a)^2\), it expands to \(x^2 - 2ax + a^2\). Recognizing these trinomials is crucial in simplifying expressions, solving equations, and understanding geometric properties. In our example, the quadratic expression \(x^2 - 14x + 49\) is a perfect square trinomial that matches the form \((x - 7)^2\). By identifying these forms, you can easily factor and simplify quadratic expressions.

To identify a perfect square trinomial, follow these steps:

• Ensure the leading coefficient (in front of \(x^2\)) is 1.
• Identify the middle term, which should be twice the product of the square roots of the first and last terms.
• Check if the last term is a perfect square whose root is used in forming the middle term.

Recognizing these patterns will allow you to transform complex expressions into manageable forms.

Quadratic Expressions

Quadratic expressions are polynomials of degree two. They take the general form: \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. These expressions are fundamental in algebra and are used to model a variety of real-world scenarios like projectile motion, area calculations, and optimization problems.

Understanding the components of a quadratic expression is essential:

• The term \(ax^2\) is the quadratic term which gives the expression its parabolic shape.
• The term \(bx\) is the linear term, influencing the symmetry and direction of the curve.
• The constant \(c\) is the y-intercept of the corresponding quadratic function.

By exploring these elements, you can learn how adjustments in the values of \(a\), \(b\), and \(c\) impact the graph and solutions of the quadratic equation. Hence, mastering quadratic expressions makes factoring, graphing, and solving these forms much more intuitive.

Factored Form

The factored form of a quadratic expression is crucial for simplifying and solving quadratic equations. It is obtained when a quadratic is expressed as a product of its binomial factors. For instance, our initial quadratic \(x^2 - 14x + 49\) is rewritten in factored form as \((x - 7)^2\), showing it is a perfect square. This conversion helps in identifying the roots (solutions) of the equation by setting each factor to zero.

The advantages of using factored form include:

• Simplifying the solution of quadratic equations by making it easy to apply the zero-product property.
• Identifying repeated roots, especially in perfect square trinomials.
• Providing insights into the behavior of quadratic functions such as symmetry.

Moving from a standard quadratic to its factored version involves recognizing factorable trinomials, using tools like the quadratic formula for more complicated expressions, or recognizing patterns in perfect square forms. Mastery over these techniques aids greatly in algebra, calculus, and even geometry.