Solving Quadratic Equations by Completing the Square: The Step-by-Step Procedure
Quadratic equations are a fundamental concept in algebra, and one of the most common methods to solve them is by completing the square. This method provides a systematic approach to transform a quadratic equation into a perfect square trinomial. Let's explore the process step-by-step using the example equation 3x^2 - 2x - 4 0.
Understanding Quadratic Equations
A quadratic equation is an equation of the form ax^2 bx c 0, where a, b, and c are constants, and a ≠ 0. This equation can be solved using various methods, including factoring, the quadratic formula, or completing the square.
Completing the Square: Step-by-Step Guide
The completing the square method involves transforming the quadratic equation into the form (x - h)^2 k. Here's a detailed step-by-step guide:
Step 1: Isolate the Terms Involving the Variable
First, we need to move the constant term to the right side of the equation.
3x^2 - 2x 4
Step 2: Divide Both Sides by the Leading Coefficient
The leading coefficient is the coefficient of the term with the highest degree (in this case, 3). We divide both sides of the equation by this coefficient to simplify the equation.
x^2 - frac{2}{3}x frac{4}{3}
Step 3: Complete the Square
To complete the square, we need to add and subtract a value on the left side of the equation such that the left side becomes a perfect square trinomial.
We find this value by taking half of the coefficient of the linear term (-2/3), squaring it, and adding it to both sides of the equation.
x^2 - frac{2}{3}x left(frac{1}{3}right)^2 frac{4}{3} left(frac{1}{3}right)^2
Adding left(frac{1}{3}right)^2 to both sides, we get:
x^2 - frac{2}{3}x frac{1}{9} frac{4}{3} frac{1}{9}
left(x - frac{1}{3}right)^2 frac{12}{9} frac{1}{9}
left(x - frac{1}{3}right)^2 frac{13}{9}
Step 4: Solve for x
Now, we take the square root of both sides of the equation, remembering to consider both the positive and negative roots.
x - frac{1}{3} pm sqrt{frac{13}{9}}
x - frac{1}{3} pm frac{sqrt{13}}{3}
Adding frac{1}{3} to both sides, we get:
x frac{1}{3} pm frac{sqrt{13}}{3}
Therefore, the solutions are:
x frac{1 sqrt{13}}{3} and x frac{1 - sqrt{13}}{3}
Generalizing the Method for All Quadratic Equations
Let's generalize the method for any quadratic equation of the form ax^2 bx c 0.
Step 1: Isolate the Constant Term
Move the constant term to the right side.
ax^2 bx -c
Step 2: Divide Both Sides by a
Divide both sides by the leading coefficient a.
x^2 frac{b}{a}x -frac{c}{a}
Step 3: Complete the Square
Add and subtract (left(frac{b}{2a}right)^2) on the left side.
x^2 frac{b}{a}x left(frac{b}{2a}right)^2 - left(frac{b}{2a}right)^2 -frac{c}{a} left(frac{b}{2a}right)^2
left(x frac{b}{2a}right)^2 - left(frac{b}{2a}right)^2 -frac{c}{a} left(frac{b}{2a}right)^2
Step 4: Solve for x
Take the square root of both sides and isolate x.
left(x frac{b}{2a}right)^2 frac{4ac - b^2}{4a^2}
x frac{b}{2a} pm sqrt{frac{4ac - b^2}{4a^2}}
x frac{b}{2a} pm frac{sqrt{4ac - b^2}}{2a}
x -frac{b}{2a} pm frac{sqrt{4ac - b^2}}{2a}
Thus, the solutions to the general quadratic equation are:
x frac{-b pm sqrt{4ac - b^2}}{2a}
Conclusion
Completing the square is a powerful technique that helps transform a quadratic equation into a perfect square trinomial. This method is not only useful for solving equations but also provides deeper insight into the structure of quadratic functions. By mastering this technique, you can solve a wide variety of quadratic equations effectively and efficiently.