Question

Use the method of completing the square to derive the formula for solving a quadratic equation.

Short answer

Answer: The steps to derive the quadratic formula are: 1) Write the general quadratic equation ax^2 + bx + c = 0. 2) Divide by a to make the coefficient of x^2 equal to 1: x^2 + (b/a)x + c/a = 0. 3) Move the constant term (c/a) to the right side of the equation: x^2 + (b/a)x = -c/a. 4) Complete the square by adding (b/2a)^2 to both sides: x^2 + (b/a)x + (b/2a)^2 = (b/2a)^2 - c/a. 5) Factor the left-hand side as the square of the binomial: (x + b/2a)^2 = (b^2 - 4ac) / 4a^2. 6) Take the square root of both sides: x + b/2a = ±√[(b^2 - 4ac) / 4a^2]. 7) Solve for x: x = [-b ± √(b^2 - 4ac)] / 2a, which is the quadratic formula.

Step-by-step solution

Write the quadratic equation

Write down the general form of a quadratic equation: ax^2 + bx + c = 0.

Divide by a

Divide both sides of the equation by a to make the coefficient of the x^2 term equal to 1: x^2 + (b/a)x + c/a = 0.

Move the constant term to the right side

Move the constant term (c/a) to the right side of the equation: x^2 + (b/a)x = -c/a.

Complete the square

Add the square of half of the coefficient of the x term, (b/2a)^2, to both sides of the equation: x^2 + (b/a)x + (b/2a)^2 = (b/2a)^2 - c/a.

Factor the left-hand side

Factor the left-hand side of the equation as the square of the binomial: (x + b/2a)^2 = (b^2 - 4ac) / 4a^2.

Take the square root of both sides

Take the square root of both sides of the equation: x + b/2a = ±√[(b^2 - 4ac) / 4a^2].

Solve for x

Solve for x by subtracting b/2a from both sides: x = [-b ± √(b^2 - 4ac)] / 2a. This is the quadratic formula, which can be used to solve any quadratic equation of the form ax^2 + bx + c = 0.

Key concepts

Completing the Square

Completing the square is a technique used to solve quadratic equations. It involves transforming a quadratic equation into a perfect square trinomial plus a constant. This method starts with the standard form of a quadratic equation: \(ax^2 + bx + c = 0\). The goal is to rearrange the equation so that the left side becomes a complete square.
Throughout this process, each step is critical: moving the constant term to the right side, dividing by the leading coefficient if it's not 1, and adding the square of half the 'b' term coefficient to both sides. They key aspect is the creation of a perfect square trinomial, which can then easily be factored.
For example, take \(x^2 + 6x + 9 = 0\), where \(6/2 = 3\) and \(3^2 = 9\). By adding and subtracting 9 inside the equation, we get \(x^2 + 6x + 9 - 9 = 0\), leading to \( (x + 3)^2 - 9 = 0 \). This simplifies to \( (x + 3)^2 = 9\), and by taking square roots, we find \(x = -3 \pm 3\), thus \(x = 0\) or \(x = -6\).
Understanding how to shape perfect square trinomials can significantly simplify solving quadratic equations.

Quadratic Formula Derivation

The quadratic formula offers a direct solution to any quadratic equation of the form \(ax^2 + bx + c = 0\). The derivation of this formula is a classic application of the completing the square method and illustrates how algebraic manipulation can reveal general solutions.
Starting with \(ax^2 + bx + c = 0\), dividing by \(a\) and then moving the term \(c/a\) to the right-side give rise to \(x^2 + (b/a)x = -c/a\). Adding \( (b/2a)^2 \) balances the equation and results in \( (x + b/2a)^2 \), which is a perfect square.
Finally, taking the square root of both sides and isolating \(x\) gives the renowned quadratic formula: \(x = [-b \pm \sqrt{b^2 - 4ac}] / 2a\). The beauty of this derivation is that it emphasizes the power of algebra to generalize solutions, setting a foundation for more complex problem-solving strategies.

Solving Quadratic Equations

Solving quadratic equations is a fundamental skill in algebra that opens the door to various mathematical concepts. A quadratic equation is characterized by its highest exponent of 2. The solutions of these equations, also known as 'roots', can be real or complex depending on the discriminant \( (b^2 - 4ac) \).
There are several methods to solve these, including factoring, graphing, the quadratic formula, and completing the square. Each method serves different purposes and contexts. For instance, factoring is efficient when the quadratic can be easily decomposed into binomial factors, while graphing provides a visual representation of the roots.
The rigorous method of solving by the quadratic formula ensures that all real and complex solutions are found, an indispensable tool for equations that cannot be factored. Successful mastery of quadratic equations equips students with the ability to approach a wide array of problems in both academic and real-world contexts.

Algebraic Methods

Algebraic methods in the context of quadratic equations encompass a range of techniques to manipulate and solve equations. These methods provide a systematic approach for tackling mathematical problems, from simple operations like addition and subtraction to more complex ones like factoring and rationalizing denominators.

• Factoring: This method is time-honored and is easiest when the quadratic equation is factorable over the integers.
• Quadratic Formula: This tool is universally applicable and guarantees a solution for any quadratic equation, whether the roots are rational or irrational, real or complex.
• Completing the Square: As a foundational algebraic method, this not only assists in solving equations but also in understanding the derivation of the quadratic formula.

Each algebraic method has its own set of rules and conditions for application, which allows for flexibility and diversity in approach. Mastery of these methods not only aids in solving equations but also in developing critical thinking and problem-solving skills, essential in the study of mathematics and science.