Solving Quadratic Equations Correcting The Completing The Square Method

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Adiya's approach to solving the quadratic equation x2+6=20xx^2 + 6 = 20x by completing the square involves dividing the constant term 6 by 2, squaring the result, and adding it to both sides. This method is incorrect for solving quadratic equations by completing the square. Completing the square is a powerful technique used to rewrite a quadratic equation in a form that allows for easy solution by isolating the variable. However, it relies on manipulating the equation in a specific way that differs significantly from Adiya's approach. In this comprehensive explanation, we will delve into the correct steps for completing the square, identify the errors in Adiya's method, and provide a detailed walkthrough of the proper solution.

The Correct Steps for Completing the Square

To accurately solve a quadratic equation of the form ax2+bx+c=0ax^2 + bx + c = 0 by completing the square, you need to follow a precise sequence of steps. These steps ensure that the equation is transformed correctly while maintaining its balance and allowing for the isolation of the variable. The process involves manipulating the equation to create a perfect square trinomial, which can then be easily factored. Here’s a breakdown of the correct methodology:

  1. Rearrange the Equation: The first crucial step is to rewrite the equation in the standard form ax2+bx+c=0ax^2 + bx + c = 0. This arrangement ensures that all terms are on one side of the equation, making it easier to identify the coefficients and constants needed for the next steps. For the given equation, x2+6=20xx^2 + 6 = 20x, this involves subtracting 20x20x from both sides to get x2−20x+6=0x^2 - 20x + 6 = 0.
  2. Isolate the Quadratic and Linear Terms: Next, isolate the terms containing x2x^2 and xx on one side of the equation. This is done by moving the constant term to the other side. In our example, this means subtracting 6 from both sides, resulting in x2−20x=−6x^2 - 20x = -6. This step sets the stage for completing the square by creating space to add a constant that will form a perfect square trinomial.
  3. Complete the Square: This is the core of the method. To complete the square, take half of the coefficient of the xx term (which is bb), square it, and add the result to both sides of the equation. The coefficient of the xx term in our rearranged equation is -20. Half of -20 is -10, and squaring -10 gives us 100. Therefore, we add 100 to both sides of the equation: x2−20x+100=−6+100x^2 - 20x + 100 = -6 + 100.
  4. Factor the Perfect Square Trinomial: The left side of the equation is now a perfect square trinomial, which can be factored into the form (x−h)2(x - h)^2, where hh is half the coefficient of the original xx term. In our case, x2−20x+100x^2 - 20x + 100 factors to (x−10)2(x - 10)^2. The equation now looks like this: (x−10)2=94(x - 10)^2 = 94.
  5. Solve for x: Take the square root of both sides of the equation. Remember to consider both positive and negative square roots. This gives us x−10=±94x - 10 = ±\sqrt{94}.
  6. Isolate x: Finally, isolate xx by adding 10 to both sides: x=10±94x = 10 ± \sqrt{94}. Thus, the solutions to the quadratic equation are x=10+94x = 10 + \sqrt{94} and x=10−94x = 10 - \sqrt{94}.

By following these steps, you can accurately complete the square and solve any quadratic equation. The method hinges on creating a perfect square trinomial, which allows for the straightforward isolation of the variable.

Identifying the Error in Adiya's Method

Adiya's method of dividing the constant term 6 by 2, squaring the result, and adding it to both sides is a misapplication of the completing the square technique. The core issue with Adiya's approach lies in the fact that it does not correctly account for the linear term (-20x) in the quadratic equation. The process of completing the square is specifically designed to manipulate the equation in such a way that a perfect square trinomial is formed, which can then be easily factored. This requires focusing on the coefficient of the linear term, not the constant term alone.

Specifically, Adiya's method involves the following steps:

  1. Divide 6 by 2, which equals 3.
  2. Square 3, which equals 9.
  3. Add 9 to both sides of the original equation.

Applying this to the equation x2+6=20xx^2 + 6 = 20x, Adiya would get:

x2+6+9=20x+9x^2 + 6 + 9 = 20x + 9

x2+15=20x+9x^2 + 15 = 20x + 9

This manipulation does not lead to a perfect square trinomial on either side of the equation. A perfect square trinomial is a trinomial that can be factored into the form (x+a)2(x + a)^2 or (x−a)2(x - a)^2. For example, x2+6x+9x^2 + 6x + 9 is a perfect square trinomial because it can be factored into (x+3)2(x + 3)^2. Adiya’s method fails to create this structure, which is essential for solving the equation by completing the square.

The fundamental error is that Adiya’s approach does not address the necessary transformation to create a perfect square. The constant that needs to be added to complete the square is derived from half the coefficient of the x term, not from the original constant term in the equation. In the equation x2−20x+6=0x^2 - 20x + 6 = 0, the coefficient of the x term is -20. It is half of this value, squared, that should be added to both sides to complete the square.

By focusing solely on the constant term, Adiya's method bypasses the crucial step of adjusting the equation to form a perfect square trinomial. This misstep prevents the equation from being factored into a manageable form, thus rendering the solution incorrect. Understanding this distinction is key to mastering the technique of completing the square and applying it effectively to solve quadratic equations.

Step-by-Step Solution Using the Correct Method

To provide a clear understanding of the correct method, let's walk through the step-by-step solution of the quadratic equation x2+6=20xx^2 + 6 = 20x by completing the square. This detailed walkthrough will highlight the critical steps and demonstrate how to correctly apply the technique.

  1. Rearrange the Equation: The first step is to rewrite the equation in the standard form ax2+bx+c=0ax^2 + bx + c = 0. This involves moving all terms to one side of the equation. Starting with x2+6=20xx^2 + 6 = 20x, we subtract 20x20x from both sides to get:

    x2−20x+6=0x^2 - 20x + 6 = 0

    This arrangement allows us to clearly identify the coefficients and constants needed for the subsequent steps.

  2. Isolate the Quadratic and Linear Terms: Next, isolate the terms containing x2x^2 and xx on one side of the equation by moving the constant term to the other side. We subtract 6 from both sides:

    x2−20x=−6x^2 - 20x = -6

    This step prepares the equation for the completion of the square by setting up the addition of a constant that will form a perfect square trinomial.

  3. Complete the Square: To complete the square, we need to add a value to both sides of the equation that will make the left side a perfect square trinomial. This value is determined by taking half of the coefficient of the xx term, squaring it, and adding the result to both sides. The coefficient of the xx term is -20. Half of -20 is -10, and squaring -10 gives us 100. Therefore, we add 100 to both sides:

    x2−20x+100=−6+100x^2 - 20x + 100 = -6 + 100

    This step is the core of the method, transforming the left side into a perfect square trinomial.

  4. Factor the Perfect Square Trinomial: The left side of the equation is now a perfect square trinomial, which can be factored into the form (x−h)2(x - h)^2, where hh is half the coefficient of the original xx term. In this case, x2−20x+100x^2 - 20x + 100 factors to (x−10)2(x - 10)^2. The right side simplifies to 94, so the equation becomes:

    (x−10)2=94(x - 10)^2 = 94

    This factored form allows us to easily solve for xx.

  5. Solve for x: Take the square root of both sides of the equation. Remember to consider both positive and negative square roots:

    (x−10)2=±94\sqrt{(x - 10)^2} = ±\sqrt{94}

    x−10=±94x - 10 = ±\sqrt{94}

    This step isolates xx further, preparing for the final solution.

  6. Isolate x: Finally, isolate xx by adding 10 to both sides:

    x=10±94x = 10 ± \sqrt{94}

    Thus, the solutions to the quadratic equation are x=10+94x = 10 + \sqrt{94} and x=10−94x = 10 - \sqrt{94}.

By following these steps, we have correctly completed the square and found the solutions to the given quadratic equation. This detailed solution underscores the importance of each step and clarifies the method for solving similar equations.

Why Completing the Square Works: The Underlying Principle

To fully grasp the method of completing the square, it's crucial to understand the underlying mathematical principle that makes it work. The technique is rooted in the algebraic identity that describes a perfect square trinomial. By manipulating a quadratic equation to fit this form, we can transform it into a more manageable equation that can be easily solved.

The fundamental principle behind completing the square is based on the algebraic identity:

(x+a)2=x2+2ax+a2(x + a)^2 = x^2 + 2ax + a^2

This identity shows that a perfect square trinomial is formed by squaring a binomial (x+a)(x + a). The trinomial consists of three terms: the square of xx (x2x^2), twice the product of xx and aa (2ax2ax), and the square of aa (a2a^2). Conversely, if we have an expression in the form x2+bxx^2 + bx, we can complete the square by adding a constant term that makes the entire expression a perfect square trinomial. This constant term is found by taking half of the coefficient of the xx term (which is bb) and squaring it.

To illustrate this, let’s consider the general quadratic expression x2+bxx^2 + bx. To complete the square, we need to add a term cc such that:

x2+bx+c=(x+a)2x^2 + bx + c = (x + a)^2

Expanding (x+a)2(x + a)^2, we get:

x2+2ax+a2x^2 + 2ax + a^2

Comparing the coefficients, we see that:

b=2ab = 2a

c=a2c = a^2

From the first equation, we can express aa in terms of bb:

a=b2a = \frac{b}{2}

Substituting this into the second equation, we find the value of cc needed to complete the square:

c=(b2)2=b24c = (\frac{b}{2})^2 = \frac{b^2}{4}

Thus, to complete the square for an expression x2+bxx^2 + bx, we add (b2)2(\frac{b}{2})^2 to it. This is the key insight that drives the method of completing the square.

In the context of solving quadratic equations, this principle is applied by manipulating the equation to create a perfect square trinomial on one side. For example, in the equation x2−20x=−6x^2 - 20x = -6, the coefficient of the xx term is -20. To complete the square, we take half of -20, which is -10, and square it, resulting in 100. Adding 100 to both sides of the equation gives:

x2−20x+100=−6+100x^2 - 20x + 100 = -6 + 100

(x−10)2=94(x - 10)^2 = 94

The left side is now a perfect square trinomial, which can be factored as (x−10)2(x - 10)^2. This transformation allows us to solve for xx by taking the square root of both sides and isolating the variable.

Understanding this underlying principle clarifies why Adiya's method is incorrect. Adiya focused on the constant term without considering the coefficient of the xx term, which is essential for creating a perfect square trinomial. The correct method involves manipulating the equation based on the coefficient of the xx term to align with the perfect square trinomial identity.

Conclusion

In conclusion, Adiya's solution method for solving the quadratic equation x2+6=20xx^2 + 6 = 20x by completing the square is incorrect. The accurate approach involves rearranging the equation, isolating the quadratic and linear terms, completing the square by adding the correct constant (derived from half the coefficient of the xx term, squared), factoring the perfect square trinomial, and solving for xx. The step-by-step solution provided demonstrates the correct application of the method, and understanding the underlying principle of perfect square trinomials clarifies why this technique works. Mastering this method allows for the effective solution of quadratic equations that may not be easily factorable by other means, providing a versatile tool in mathematical problem-solving.