Problem 76 - Introductory Algebra for College Students

Question

Any quadratic equation that can be solved by completing the square can be solved by the quadratic formula.

Step-by-Step Solution

Answer

The quadratic formula, 'x = (-b ± sqrt(b^2 - 4ac))/2a' is derived from the process of completing the square, and is used to find the roots or solutions of any quadratic equation.

Step 1: Understanding the Quadratic Equation

Start with a standard quadratic equation of the form 'ax^2 + bx + c = 0'. The quadratic formula '+sqrt(b^2 - 4ac) / 2a' is used to solve this equation. Here, 'a' is the coefficient of 'x^2', 'b' is the coefficient of 'x', and 'c' is the constant term.

Step 2: Finding the Quadratic Formula

To derive the quadratic formula through completing the square, first rewrite the quadratic equation as 'ax^2 + bx = -c'. Then, add 'b^2 / (4a^2)' to both sides to complete the square, which gives the equation: '(x + b / 2a)^2 = (b^2 - 4ac) / (4a^2)'. Taking the square root of both sides and isolating 'x', the equation becomes 'x = -b ± sqrt(b^2 - 4ac) / 2a'.

Step 3: Solving the Quadratic Equation

Plug the coefficients 'a', 'b' and 'c' of the quadratic equation into the quadratic formula to find the roots or solutions. The solutions will be two values for 'x' that satisfy the original quadratic equation, giving 'x = (-b ± sqrt(b^2 - 4ac))/2a'.

Key Concepts

Completing the Square

Whenever you face a quadratic equation, completing the square is a methodical way to find the equation's roots. It involves reshaping the equation into a perfect square trinomial, which helps to isolate the variable.

For example, let's consider the general form of a quadratic equation, which is \( ax^2 + bx + c = 0 \) where \( a \) is not equal to zero. To complete the square, we need to follow several steps:

• Divide through by \( a \) if it is not 1, to get \( x^2 + (b/a)x = -c/a \)
• Add \( (b/2a)^2 \) to both sides to form a perfect square on the left side
• This transforms the equation into \( x^2 + (b/a)x + (b/2a)^2 = (b/2a)^2 - c/a \)
• Now, we can easily take the square root of both sides, which gives \( x + b/2a = \pm\sqrt{(b/2a)^2 - c/a} \) and solve for \( x \)

This technique is fundamental in algebra because it not only aids in solving quadratic equations but also in understanding the nature of their roots. It's a stepping stone to deriving the quadratic formula, which provides a more direct solution path.

Quadratic Equation

The quadratic equation is a fundamental concept in algebra that dictates the relationship between a polynomial and its roots. In its simplest form, a quadratic equation is written as \( ax^2 + bx + c = 0 \) where \( a \) is the coefficient of the squared term, \( b \) is the coefficient of the linear term, and \( c \) is the constant.

A key characteristic of quadratic equations is their curve-shape graph, known as a parabola. Depending on the values of \( a \) , \( b \) and \( c \) , the parabola opens upwards or downwards and may intersect the \( x \) -axis at two points, one point, or no point at all—indicating two real solutions, one real solution, or two complex solutions, respectively.

Understanding the structure of a quadratic equation is crucial for figuring out how to approach its solution. Whether using the quadratic formula, completing the square, or factoring, recognizing this form is the first step in solving quadratic equations.

Solving Quadratic Equations

Solving quadratic equations is a core skill in algebra that opens the door to understanding various mathematical concepts. There are several methods to find the solutions, also known as roots, of a quadratic equation.

• Using the quadratic formula \( x = (-b \pm \sqrt{b^2 - 4ac})/2a \) is one of the most common methods. It's a straightforward way to compute the roots for any quadratic equation of the form \( ax^2 + bx + c = 0 \).
• Factoring is another method used for solving quadratics, especially when the equation is easily decomposable into a product of binomials.
• Graphically, you can find the solutions by identifying the points where the parabola intersects the \( x \) -axis.

Each method has its advantages depending on the specific equation you're working with. For instance, completing the square is very useful when the equation does not factor neatly, while the quadratic formula can handle any form of quadratic equation. Understanding these methods enhances your problem-solving skills and helps you choose the most efficient strategy for any given problem.