Completing The Square In Quadratic Equations Finding The Missing Number For X^2 + 12x = 11

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Completing the square is a powerful technique in algebra that allows us to rewrite quadratic equations into a more manageable form, primarily to solve for the unknown variable, x. The method involves transforming a quadratic expression into a perfect square trinomial, which can then be easily factored and solved. In this comprehensive guide, we will delve into the intricacies of completing the square, specifically focusing on the equation x2+12x=11x^2 + 12x = 11. Our main objective is to determine the precise number that needs to be added to both sides of the equation to successfully complete the square. By understanding the underlying principles and applying the correct steps, you will be able to confidently tackle similar problems and gain a deeper understanding of quadratic equations.

Understanding Perfect Square Trinomials

Before we dive into the specifics of our equation, it's crucial to understand the concept of perfect square trinomials. Perfect square trinomials are quadratic expressions that can be factored into the square of a binomial. They follow a specific pattern, which is essential to recognize when completing the square. A perfect square trinomial can be expressed in one of two forms:

  1. (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2
  2. (aβˆ’b)2=a2βˆ’2ab+b2(a - b)^2 = a^2 - 2ab + b^2

Notice the relationship between the coefficients in these expressions. The constant term (b2b^2) is the square of half the coefficient of the linear term (2ab). This relationship is the key to completing the square. When we have an expression in the form x2+bxx^2 + bx, we need to find the constant term that, when added, will make the expression a perfect square trinomial. This constant term is calculated by taking half of the coefficient of the x term (which is b), squaring it, and then adding it to the expression. This process ensures that the resulting trinomial can be factored into the form (x + rac{b}{2})^2 or (x - rac{b}{2})^2.

For instance, let's consider the expression x2+6xx^2 + 6x. To complete the square, we take half of the coefficient of x (which is 6), which gives us 3. Squaring this value, we get 32=93^2 = 9. Adding 9 to the original expression, we get x2+6x+9x^2 + 6x + 9, which is a perfect square trinomial because it can be factored as (x+3)2(x + 3)^2. Understanding this principle is fundamental to successfully completing the square in any quadratic equation.

Identifying the Coefficient of the x Term

In our given equation, x2+12x=11x^2 + 12x = 11, the first crucial step is to identify the coefficient of the x term. The coefficient of the x term is the number that is multiplied by x. In this case, the coefficient is 12. This number plays a pivotal role in determining the value we need to add to both sides of the equation to complete the square. Once we've identified the coefficient, we can proceed to the next step, which involves calculating half of this coefficient. This calculation is essential because it directly relates to the value that will be squared to complete the perfect square trinomial. By accurately identifying the coefficient of the x term, we set the foundation for the subsequent steps in the completing the square process, ensuring that we arrive at the correct value to add and successfully transform the equation.

Calculating Half of the Coefficient

Now that we've identified the coefficient of the x term as 12, the next crucial step is to calculate half of this coefficient. This calculation is a fundamental part of the completing the square process. To find half of the coefficient, we simply divide it by 2. In our case, we divide 12 by 2, which gives us 6. This value, 6, represents the number that will be squared in the subsequent step. It is essential to understand that this value is not the final answer to our question but rather an intermediate result that plays a critical role in determining the constant term needed to complete the square. This intermediate value represents half of the coefficient of the x term and will be used to create the perfect square trinomial. By carefully calculating half of the coefficient, we are setting ourselves up for the final step of finding the number to add to both sides of the equation.

Squaring the Result

Having calculated half of the coefficient of the x term (which is 6), the next essential step is to square this result. Squaring the result means multiplying the value by itself. In our case, we need to calculate 626^2, which is 6imes6=366 imes 6 = 36. This value, 36, is the magic number we've been searching for. It's the number that, when added to both sides of the equation x2+12x=11x^2 + 12x = 11, will complete the square. Squaring the result is a critical step because it ensures that the trinomial we create on the left side of the equation will be a perfect square trinomial, meaning it can be factored into the square of a binomial. This is the very essence of the completing the square technique. By accurately squaring the result, we determine the constant term needed to transform the quadratic expression into a perfect square, allowing us to proceed with solving the equation more easily.

Adding the Number to Both Sides

Now that we've determined the number 36 by squaring half of the coefficient of the x term, the next crucial step is to add this number to both sides of the original equation, x2+12x=11x^2 + 12x = 11. Adding the same number to both sides of an equation is a fundamental principle of algebra, ensuring that the equation remains balanced and the equality is maintained. This step is the heart of the completing the square technique. When we add 36 to both sides, we get:

x2+12x+36=11+36x^2 + 12x + 36 = 11 + 36

This transformation is significant because the left side of the equation, x2+12x+36x^2 + 12x + 36, is now a perfect square trinomial. It can be factored into (x+6)2(x + 6)^2. The right side of the equation simplifies to 47. Our equation now looks like this:

(x+6)2=47(x + 6)^2 = 47

This new form is much easier to work with. We've successfully completed the square, and now we can solve for x by taking the square root of both sides and isolating x. Adding the number to both sides is the pivotal step that allows us to rewrite the equation in a more manageable form, making it possible to find the solutions for the unknown variable.

Factoring the Perfect Square Trinomial

After adding 36 to both sides of the equation, we transformed the left side into a perfect square trinomial: x2+12x+36x^2 + 12x + 36. Factoring this trinomial is the next critical step in completing the square. A perfect square trinomial, by definition, can be factored into the square of a binomial. In this case, the trinomial x2+12x+36x^2 + 12x + 36 factors into (x+6)2(x + 6)^2. This factorization is based on the pattern of perfect square trinomials, where the trinomial of the form a2+2ab+b2a^2 + 2ab + b^2 can be factored as (a+b)2(a + b)^2. In our case, a is x and b is 6, so the factored form is indeed (x+6)2(x + 6)^2. The ability to recognize and factor perfect square trinomials is a fundamental skill in algebra. It allows us to simplify complex expressions and solve quadratic equations more efficiently. By factoring the perfect square trinomial, we rewrite the equation in a form that is easier to solve, bringing us closer to finding the values of x that satisfy the equation.

Solving for x

With the equation now in the form (x+6)2=47(x + 6)^2 = 47, we're in the final stage of solving for x. Solving for x involves isolating the variable by performing inverse operations. The first step is to take the square root of both sides of the equation. Remember that when taking the square root, we must consider both the positive and negative roots:

(x+6)2=Β±47\sqrt{(x + 6)^2} = \pm\sqrt{47}

This simplifies to:

x+6=Β±47x + 6 = \pm\sqrt{47}

Next, we isolate x by subtracting 6 from both sides:

x=βˆ’6Β±47x = -6 \pm \sqrt{47}

This gives us two solutions for x:

  1. x=βˆ’6+47x = -6 + \sqrt{47}
  2. x=βˆ’6βˆ’47x = -6 - \sqrt{47}

These are the exact solutions for x. If we need approximate values, we can use a calculator to find the square root of 47, which is approximately 6.856. Therefore, the approximate solutions are:

  1. xβ‰ˆβˆ’6+6.856β‰ˆ0.856x \approx -6 + 6.856 \approx 0.856
  2. xβ‰ˆβˆ’6βˆ’6.856β‰ˆβˆ’12.856x \approx -6 - 6.856 \approx -12.856

By following the steps of completing the square, we've successfully transformed the original equation, factored the perfect square trinomial, and isolated x to find its values. This process demonstrates the power and versatility of the completing the square technique in solving quadratic equations.

Conclusion

In conclusion, to complete the square for the equation x2+12x=11x^2 + 12x = 11, the number that should be added to both sides is 36. This value was determined by taking half of the coefficient of the x term (which is 12), squaring the result (6), and obtaining 36. Adding 36 to both sides transforms the left side of the equation into a perfect square trinomial, which can then be factored and used to solve for x. Completing the square is a valuable technique in algebra for solving quadratic equations and understanding the properties of parabolas. By mastering this method, you gain a powerful tool for tackling a wide range of mathematical problems. The process involves understanding perfect square trinomials, identifying the coefficient of the x term, calculating half of it, squaring the result, adding it to both sides of the equation, factoring the trinomial, and finally, solving for x. Through these steps, we successfully navigated the equation x2+12x=11x^2 + 12x = 11 and found the number needed to complete the square, illustrating the effectiveness of this algebraic technique. Understanding and applying the completing the square method not only helps in solving equations but also provides a deeper insight into the structure and behavior of quadratic expressions and their graphical representations.