Solving Quadratic Equations X^2-10x+21=0 A Step By Step Guide

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In this comprehensive guide, we will delve into the step-by-step process of solving the quadratic equation x^2 - 10x + 21 = 0. Quadratic equations are a fundamental concept in algebra, appearing in various mathematical and real-world applications. Mastering the techniques to solve them is crucial for anyone studying mathematics, engineering, or related fields. This article provides a detailed explanation of the factoring method, a widely used technique for solving quadratic equations. We will break down each step, ensuring a clear understanding of the underlying principles. Additionally, we will explore alternative methods such as completing the square and the quadratic formula, offering a holistic approach to tackling quadratic equations. Whether you are a student looking to improve your algebra skills or a professional seeking a refresher, this guide will equip you with the knowledge and confidence to solve quadratic equations effectively.

Before we dive into solving the equation x^2 - 10x + 21 = 0, let's first understand what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. The solutions to a quadratic equation are called its roots or zeros, which represent the values of x that satisfy the equation. These roots can be real or complex numbers.

In our specific equation, x^2 - 10x + 21 = 0, we can identify the coefficients as follows: a = 1, b = -10, and c = 21. Understanding these coefficients is essential for applying various methods to solve the equation. The coefficient a determines the shape of the parabola represented by the quadratic equation, while b and c influence its position on the coordinate plane. Solving a quadratic equation means finding the x-intercepts of this parabola, where the graph crosses the x-axis. These x-intercepts are the solutions to the equation.

Quadratic equations have a rich history, with early examples found in ancient Babylonian and Egyptian texts. Over the centuries, mathematicians have developed numerous techniques to solve them, each with its own advantages and limitations. The factoring method, which we will focus on in this article, is particularly useful when the quadratic equation can be easily factored. However, other methods, such as completing the square and the quadratic formula, are more versatile and can be applied to any quadratic equation, regardless of whether it can be factored easily. By mastering these different methods, you can approach quadratic equations with confidence and choose the most efficient technique for each specific problem.

Factoring is a powerful technique for solving quadratic equations, especially when the equation can be expressed as a product of two binomials. The goal of factoring is to rewrite the quadratic expression x^2 - 10x + 21 in the form (x - p)(x - q), where p and q are constants. If we can find such p and q, then the solutions to the equation are simply x = p and x = q. This method relies on the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero.

To factor the quadratic expression x^2 - 10x + 21, we need to find two numbers that multiply to c (which is 21) and add up to b (which is -10). Let's consider the factors of 21: 1 and 21, 3 and 7. Since the sum of the two numbers must be -10, we need to consider negative factors. The pair -3 and -7 satisfy both conditions: (-3) * (-7) = 21 and (-3) + (-7) = -10. Therefore, we can rewrite the quadratic expression as (x - 3)(x - 7).

Now, we set each factor equal to zero and solve for x:

  • x - 3 = 0 implies x = 3
  • x - 7 = 0 implies x = 7

Thus, the solutions to the quadratic equation x^2 - 10x + 21 = 0 are x = 3 and x = 7. Factoring is an efficient method when the coefficients are integers and the roots are rational numbers. However, not all quadratic equations can be easily factored. In such cases, other methods like completing the square or the quadratic formula may be more appropriate. Understanding the factoring method is a crucial first step in mastering quadratic equation solving techniques.

Let's break down the process of solving the quadratic equation x^2 - 10x + 21 = 0 by factoring into a series of clear, manageable steps. This step-by-step approach will not only help you understand the solution to this particular problem but also provide a framework for solving other quadratic equations by factoring.

Step 1: Identify the Coefficients The first step is to identify the coefficients a, b, and c in the quadratic equation ax^2 + bx + c = 0. In our equation, x^2 - 10x + 21 = 0, we have:

  • a = 1 (the coefficient of x^2)
  • b = -10 (the coefficient of x)
  • c = 21 (the constant term)

Identifying these coefficients is crucial for determining the appropriate method to solve the equation and for correctly applying the factoring process.

Step 2: Find Two Numbers The next step is to find two numbers that multiply to c (21) and add up to b (-10). This is the core of the factoring method. We need to think of pairs of factors of 21 and see which pair adds up to -10. Let's list the factor pairs of 21:

  • 1 and 21
  • 3 and 7

Since b is negative, we should consider negative factors as well:

  • -1 and -21
  • -3 and -7

We see that -3 and -7 satisfy both conditions: (-3) * (-7) = 21 and (-3) + (-7) = -10. These are the numbers we need.

Step 3: Rewrite the Quadratic Equation Now, rewrite the quadratic equation by replacing the middle term (-10x) with the sum of the terms using the two numbers we found (-3x and -7x). So, x^2 - 10x + 21 = 0 becomes x^2 - 3x - 7x + 21 = 0.

Step 4: Factor by Grouping Next, we factor by grouping. Group the first two terms and the last two terms:

  • (x^2 - 3x) + (-7x + 21) = 0

Factor out the greatest common factor (GCF) from each group:

  • x(x - 3) - 7(x - 3) = 0

Notice that (x - 3) is a common factor in both terms. Factor out (x - 3):

  • (x - 3)(x - 7) = 0

Step 5: Apply the Zero-Product Property The zero-product property states that if the product of two factors is zero, then at least one of the factors must be zero. So, we set each factor equal to zero:

  • x - 3 = 0 or x - 7 = 0

Step 6: Solve for x Finally, solve each equation for x:

  • x - 3 = 0 gives x = 3
  • x - 7 = 0 gives x = 7

Therefore, the solutions to the quadratic equation x^2 - 10x + 21 = 0 are x = 3 and x = 7. This step-by-step process demonstrates how to solve a quadratic equation by factoring, providing a clear and structured approach for similar problems.

While factoring is an efficient method for solving quadratic equations, it is not always the most straightforward approach, especially when the equation is not easily factorable. Fortunately, there are alternative methods that can be used to solve quadratic equations, such as completing the square and using the quadratic formula. Understanding these methods provides a more comprehensive toolkit for tackling a wide range of quadratic equations. Let's explore these alternative techniques.

Completing the Square

Completing the square is a method that transforms a quadratic equation into a perfect square trinomial, which can then be easily solved. This method is particularly useful when the quadratic equation cannot be factored using integers. The general idea behind completing the square is to manipulate the equation ax^2 + bx + c = 0 into the form (x - h)^2 = k, where h and k are constants. Once in this form, we can simply take the square root of both sides and solve for x.

To complete the square for the equation x^2 - 10x + 21 = 0, we first move the constant term to the right side of the equation:

  • x^2 - 10x = -21

Next, we need to add a value to both sides of the equation to make the left side a perfect square trinomial. This value is calculated as (b/2)^2, where b is the coefficient of the x term. In our case, b = -10, so the value to add is (-10/2)^2 = 25. Adding 25 to both sides, we get:

  • x^2 - 10x + 25 = -21 + 25
  • x^2 - 10x + 25 = 4

Now, the left side is a perfect square trinomial, which can be factored as (x - 5)^2:

  • (x - 5)^2 = 4

Taking the square root of both sides, we get:

  • x - 5 = ±2

Solving for x, we have two equations:

  • x - 5 = 2 implies x = 7
  • x - 5 = -2 implies x = 3

Thus, the solutions are x = 3 and x = 7, which are the same solutions we found by factoring. Completing the square is a reliable method that works for any quadratic equation, even those that are difficult to factor.

The Quadratic Formula

The quadratic formula is a universal method for solving quadratic equations. It provides a direct solution for x in terms of the coefficients a, b, and c of the quadratic equation ax^2 + bx + c = 0. The quadratic formula is given by:

  • x = (-b ± √(b^2 - 4ac)) / (2a)

This formula can be applied to any quadratic equation, regardless of whether it can be factored or completed the square easily. The expression inside the square root, b^2 - 4ac, is called the discriminant. The discriminant provides information about the nature of the roots:

  • If b^2 - 4ac > 0, the equation has two distinct real roots.
  • If b^2 - 4ac = 0, the equation has one real root (a repeated root).
  • If b^2 - 4ac < 0, the equation has two complex roots.

For our equation, x^2 - 10x + 21 = 0, we have a = 1, b = -10, and c = 21. Plugging these values into the quadratic formula, we get:

  • x = (10 ± √((-10)^2 - 4 * 1 * 21)) / (2 * 1)
  • x = (10 ± √(100 - 84)) / 2
  • x = (10 ± √16) / 2
  • x = (10 ± 4) / 2

This gives us two solutions:

  • x = (10 + 4) / 2 = 14 / 2 = 7
  • x = (10 - 4) / 2 = 6 / 2 = 3

Again, we find the solutions x = 3 and x = 7, consistent with the factoring and completing the square methods. The quadratic formula is a powerful tool that ensures you can solve any quadratic equation, making it an essential technique in algebra.

In conclusion, we have successfully solved the quadratic equation x^2 - 10x + 21 = 0 using the factoring method and verified our results with alternative methods such as completing the square and the quadratic formula. The solutions we found are x = 3 and x = 7. Mastering these techniques is crucial for solving a wide variety of quadratic equations and applying them to real-world problems. Whether you prefer the efficiency of factoring, the reliability of completing the square, or the universality of the quadratic formula, having a solid understanding of these methods will empower you to tackle any quadratic equation with confidence. Remember to practice regularly and explore different types of problems to further enhance your skills in algebra.