Solving $d^2 - 20d + 100 = 81$ Using The Square Root Property

In the realm of algebra, quadratic equations hold a significant position, appearing in various mathematical and scientific contexts. Among the methods employed to solve these equations, the square root property stands out for its elegance and efficiency, particularly when dealing with equations in a specific form. This article delves into the application of the square root property, providing a step-by-step guide to solving quadratic equations, complete with examples and explanations. We will specifically address the equation $d^2 - 20d + 100 = 81$, showcasing how the square root property can be effectively utilized to find its solutions.

Understanding Quadratic Equations

Before diving into the intricacies of the square root property, it is crucial to establish a firm understanding of quadratic equations. A quadratic equation is a polynomial equation of the second degree, meaning that the highest power of the variable is 2. The general form of a quadratic equation is expressed as:

$ax^2 + bx + c = 0$

where 'a', 'b', and 'c' are constants, and 'x' represents the variable. The coefficients 'a', 'b', and 'c' play a pivotal role in determining the nature and number of solutions to the equation. When a quadratic equation is presented in the form of a perfect square, the square root property becomes a powerful tool for finding its solutions.

The Square Root Property: A Powerful Tool

The square root property is a mathematical principle that states: if $x^2 = k$, where k is a non-negative real number, then $x = \sqrt{k}$ or $x = -\sqrt{k}$. This property arises directly from the definition of the square root. The square root of a number 'k' is a value that, when multiplied by itself, equals 'k'. Consequently, any non-negative number has two square roots: a positive square root and a negative square root.

The square root property provides a direct method for solving quadratic equations that can be expressed in the form $(x + a)^2 = k$ or $(x - a)^2 = k$, where 'a' and 'k' are constants. This form represents a perfect square trinomial, which is a quadratic expression that can be factored into the square of a binomial.

Solving $d^2 - 20d + 100 = 81$ Using the Square Root Property

Let's apply the square root property to solve the quadratic equation $d^2 - 20d + 100 = 81$. This equation presents an excellent opportunity to demonstrate the effectiveness of this method.

Step 1: Recognize the Perfect Square Trinomial

The first crucial step is to recognize that the left-hand side of the equation, $d^2 - 20d + 100$, is a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the square of a binomial. In this case, we can factor the expression as follows:

$d^2 - 20d + 100 = (d - 10)^2$

This factorization is achieved by recognizing that the first and last terms are perfect squares ($d^2$ and $100 = 10^2$), and the middle term is twice the product of the square roots of the first and last terms ($-20d = -2 * d * 10$).

Step 2: Rewrite the Equation

Now that we have factored the left-hand side, we can rewrite the original equation as:

$(d - 10)^2 = 81$

This rewritten form clearly showcases the structure required for applying the square root property.

Step 3: Apply the Square Root Property

With the equation in the form $(d - 10)^2 = 81$, we can now apply the square root property. Taking the square root of both sides of the equation, we get:

$\sqrt{(d - 10)^2} = \pm \sqrt{81}$

This simplifies to:

$d - 10 = \pm 9$

Remember that we include both the positive and negative square roots of 81, which are 9 and -9, respectively. This is a crucial step in ensuring we find all possible solutions to the quadratic equation.

Step 4: Solve for 'd'

The equation $d - 10 = \pm 9$ represents two separate linear equations:

1. $d - 10 = 9$
2. $d - 10 = -9$

Solving the first equation, we add 10 to both sides:

$d = 9 + 10$

$d = 19$

Solving the second equation, we also add 10 to both sides:

$d = -9 + 10$

$d = 1$

Therefore, the solutions to the quadratic equation $d^2 - 20d + 100 = 81$ are $d = 19$ and $d = 1$.

Step 5: Verify the Solutions (Optional but Recommended)

To ensure the accuracy of our solutions, we can substitute each value of 'd' back into the original equation and verify that it holds true.

For $d = 19$:

$(19)^2 - 20(19) + 100 = 361 - 380 + 100 = 81$

For $d = 1$:

$(1)^2 - 20(1) + 100 = 1 - 20 + 100 = 81$

Both solutions satisfy the original equation, confirming their validity.

Expressing the Solution

The solutions to the quadratic equation $d^2 - 20d + 100 = 81$ are $d = 19$ and $d = 1$. Following the requested format, we list the solutions separated by a comma:

$d = 1, 19$

When to Use the Square Root Property

The square root property is most effectively applied when the quadratic equation can be easily expressed in the form $(x + a)^2 = k$ or $(x - a)^2 = k$. This typically occurs when the quadratic equation is a perfect square trinomial or can be manipulated into that form. While other methods, such as factoring and the quadratic formula, can be used to solve any quadratic equation, the square root property provides a more direct and efficient approach in specific cases.

Key scenarios where the square root property is advantageous:

• When the quadratic equation lacks a linear term (i.e., the 'b' coefficient is 0). For example, $x^2 - 9 = 0$ can be easily solved using the square root property.
• When the quadratic equation is already in the form of a perfect square trinomial. As demonstrated in the example above, $d^2 - 20d + 100 = 81$ readily lends itself to the square root property.

Alternative Methods for Solving Quadratic Equations

While the square root property is a valuable tool, it is essential to be aware of other methods for solving quadratic equations. These methods provide alternative approaches and can be particularly useful when the square root property is not directly applicable.

1. Factoring

Factoring is a technique that involves expressing the quadratic equation as a product of two linear factors. For example, the equation $x^2 + 5x + 6 = 0$ can be factored as $(x + 2)(x + 3) = 0$. Setting each factor equal to zero yields the solutions $x = -2$ and $x = -3$. Factoring is most effective when the coefficients of the quadratic equation are integers and the roots are rational numbers.

2. The Quadratic Formula

The quadratic formula is a universal method for solving any quadratic equation, regardless of the nature of its coefficients or roots. The formula is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where 'a', 'b', and 'c' are the coefficients of the quadratic equation $ax^2 + bx + c = 0$. The quadratic formula provides the solutions directly, even when factoring is difficult or impossible. However, it can be more computationally intensive than other methods.

3. Completing the Square

Completing the square is a technique that transforms a quadratic equation into a perfect square trinomial, allowing it to be solved using the square root property. This method involves manipulating the equation algebraically to create a perfect square on one side. Completing the square is a versatile method but can be more complex than other techniques in certain cases.

Conclusion

The square root property offers a straightforward and efficient method for solving quadratic equations that can be expressed in the form $(x + a)^2 = k$ or $(x - a)^2 = k$. By recognizing perfect square trinomials and applying the square root property, we can readily find the solutions to these equations. In the case of $d^2 - 20d + 100 = 81$, we successfully applied the square root property to determine the solutions $d = 1$ and $d = 19$. While the square root property is not universally applicable to all quadratic equations, it remains a valuable tool in the algebraic toolbox, particularly when combined with other methods such as factoring, the quadratic formula, and completing the square. A comprehensive understanding of these techniques empowers mathematicians and problem-solvers to tackle a wide range of quadratic equations with confidence and precision.

Remember to always verify your solutions by substituting them back into the original equation. This practice helps to ensure accuracy and reinforces your understanding of the problem-solving process.