Problem 25 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 25-26: Determine the value or values of \(d\) that complete the square for the expression. (a) \(x^{2}+9 x+d\) (b) \(x^{2}-8 x+d\) (c) \(x^{2}+d x+36\) (d) \(x^{2}+d x+\frac{49}{4}\)

Step-by-Step Solution

Answer

(a) 20.25, (b) 16, (c) 12 or -12, (d) 7 or -7.

Step 1: Understand Completing the Square

To complete the square for an expression of the form \(x^2 + bx + c\), we aim to transform it into \((x + m)^2\). The relationship is such that \(b = 2m\), so \(m = \frac{b}{2}\). The term \(d\) is \(m^2\), which is the missing number needed.

Step 2: Solve Part (a)

For the expression \(x^2 + 9x + d\), identify \(b\) as 9. Then \(m = \frac{9}{2} = 4.5\). Therefore, \(d\) is \(m^2 = (4.5)^2 = 20.25\).

Step 3: Solve Part (b)

For the expression \(x^2 - 8x + d\), identify \(b\) as -8. Then \(m = \frac{-8}{2} = -4\). Therefore, \(d\) is \(m^2 = (-4)^2 = 16\).

Step 4: Solve Part (c)

For the expression \(x^2 + dx + 36\), we want \((x + m)^2\) to equal \(x^2 + dx + 36\). Given \(m^2 = 36\), \(m = 6\) or \(m = -6\). Thus, \(d = 2m = 12\) or \(d = -12\).

Step 5: Solve Part (d)

For the expression \(x^2 + dx + \frac{49}{4}\), we want \((x + m)^2\) to equal \(x^2 + dx + \frac{49}{4}\). Given \(m^2 = \frac{49}{4}\), \(m = \frac{7}{2}\) or \(m = -\frac{7}{2}\). Thus, \(d = 2m = 7\) or \(d = -7\).

Key Concepts

Quadratic Expressions

Quadratic expressions are a foundational concept in algebra. They typically include terms of the form \(x^2 + bx + c\). These equations are used in numerous areas, like physics, engineering, and economics, because they model parabolic motion and can represent relationships between variables.
The standard form of a quadratic expression is \(ax^2 + bx + c\), but when completing the square, we focus on cases where \(a = 1\), simplifying it to \(x^2 + bx + c\). Understanding quadratic expressions is crucial since they lay the groundwork for solving quadratic equations, which is a key skill in algebra. By grasping the structure and behavior of these equations, students can tackle complex problems more effectively.

• Parabolic nature: The graph of a quadratic expression is a curve called a parabola.
• Zero points: Quadratic equations can have up to two solutions, known as roots or x-intercepts.
• Symmetrical property: Quadratics are symmetrical around a vertical line through their vertex.

Mathematical Concepts

Completing the square is one of the essential mathematical concepts in algebra. This technique manipulates a quadratic expression to take the form of a perfect square trinomial. In other words, it rewrites the expression as \((x + m)^2\). This transformation simplifies solving and analyzing quadratics.
The key lies in understanding and manipulating the middle term \(bx\). By identifying the appropriate value for \(m\), this technique helps to break down any quadratic expression into a format that is easily comparable and solvable. This is particularly useful for solving quadratic equations or converting equations into vertex form.

• Connection to quadratic formula: Completing the square is a stepping stone to deriving the quadratic formula.
• Advantages: It provides a clear path to find the roots of the equation and determine the vertex.
• Examples: Because of its flexibility, completing the square can solve real-world problems that require quadratic modeling.

Algebraic Techniques

Algebraic techniques like completing the square are fundamental for manipulating expressions and equations in different branches of mathematics. This technique involves converting a quadratic expression into a perfect square trinomial to make it easier to work with.
The process is straightforward: identify the next number \(d\) needed to create a perfect square trinomial from \(x^2 + bx + d\). By calculating \(m = \frac{b}{2}\) and then \(d = m^2\), we can complete the square. Let's walk through it:

• Identify \(b\): Begin by identifying the coefficient of \(x\).
• Calculate \(m\): Divide \(b\) by 2 to find \(m\).
• Compute \(d\): Square \(m\) to find the value of \(d\) that completes the square.

This technique allows for both simplification and deeper analysis of quadratic expressions. It shifts the perspective from focusing solely on individual terms to understanding the expression as a whole, fostering a more intuitive grasp of algebra.