Solving Quadratic Equations: A Step-by-Step Guide

by Alex Johnson 50 views

Let's dive into solving quadratic equations! Quadratic equations are polynomial equations of the second degree. They have the general form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'x' is the variable we want to find. Today, we're tackling the specific equation −x2+8x+10=0-x^2 + 8x + 10 = 0. Buckle up; we'll explore different methods to crack this mathematical nut!

Understanding Quadratic Equations

Before we jump into solving, let's build a solid foundation. A quadratic equation is essentially a polynomial equation where the highest power of the variable is 2. This means we're dealing with something that can be expressed as ax² + bx + c = 0. The coefficients 'a', 'b', and 'c' play crucial roles in determining the nature and solutions of the equation. The solutions, also known as roots, are the values of 'x' that satisfy the equation, making the left-hand side equal to zero. These roots represent the points where the parabola defined by the quadratic equation intersects the x-axis on a graph. Understanding the structure of a quadratic equation is the first step in mastering the techniques to solve them. Recognizing the coefficients and their impact on the equation's behavior allows for a more intuitive approach to problem-solving. For example, the sign of 'a' determines whether the parabola opens upwards or downwards, influencing the existence and nature of real solutions. Moreover, the discriminant, calculated as b² - 4ac, provides valuable information about the number and type of roots. A positive discriminant indicates two distinct real roots, a zero discriminant implies one real root (a repeated root), and a negative discriminant suggests two complex roots. Therefore, grasping these fundamental concepts provides a robust framework for tackling a wide range of quadratic equations with confidence and precision. Analyzing the equation before attempting to solve it can often lead to a more efficient and insightful solution process. Whether you are a student learning algebra or a professional applying mathematical models, a deep understanding of quadratic equations is an invaluable asset.

Method 1: Using the Quadratic Formula

The quadratic formula is your go-to tool for solving any quadratic equation. It's a universal key that unlocks the solutions regardless of the complexity of the equation. The formula is expressed as: x = (-b ± √(b² - 4ac)) / 2a. Let's break down how to apply this to our equation, −x2+8x+10=0-x^2 + 8x + 10 = 0. First, identify 'a', 'b', and 'c'. In this case, a = -1, b = 8, and c = 10. Now, plug these values into the formula: x = (-8 ± √(8² - 4(-1)(10))) / (2 * -1). Simplify the expression under the square root: 8² - 4(-1)(10) = 64 + 40 = 104. So, the equation becomes: x = (-8 ± √104) / -2. We can simplify √104 by factoring out the largest perfect square, which is 4: √104 = √(4 * 26) = 2√26. Now, the equation looks like: x = (-8 ± 2√26) / -2. Finally, divide both terms in the numerator by -2: x = 4 ± √26. Thus, we have two solutions: x = 4 + √26 and x = 4 - √26. These are the exact solutions to the quadratic equation. Approximating the value of √26 (which is roughly 5.1), we find the approximate solutions: x ≈ 4 + 5.1 ≈ 9.1 and x ≈ 4 - 5.1 ≈ -1.1. The quadratic formula ensures that you can solve any quadratic equation, regardless of its factorability or complexity. It provides a straightforward, algorithmic approach, making it an indispensable tool in algebra. The formula is derived by completing the square in the general quadratic equation, providing a powerful and reliable method for finding the roots.

Step-by-Step Solution with the Quadratic Formula

Let’s walk through applying the quadratic formula to the equation −x2+8x+10=0-x^2 + 8x + 10 = 0 step-by-step, making sure every detail is crystal clear. The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a.

  1. Identify a, b, and c:

    • In our equation, a = -1, b = 8, and c = 10.

  2. Plug the values into the formula:

    • x = (-8 ± √(8² - 4(-1)(10))) / (2 * -1)

  3. Simplify the expression under the square root (the discriminant):

    • 8² - 4(-1)(10) = 64 + 40 = 104

  4. Rewrite the equation with the simplified discriminant:

    • x = (-8 ± √104) / -2

  5. Simplify the square root (if possible):

    • √104 can be simplified by factoring out the largest perfect square: √104 = √(4 * 26) = 2√26

  6. Substitute the simplified square root back into the equation:

    • x = (-8 ± 2√26) / -2

  7. Divide both terms in the numerator by the denominator:

    • x = 4 ± √26

  8. State the two solutions:

    • x = 4 + √26 and x = 4 - √26

  9. Approximate the solutions (if needed):

    • Since √26 ≈ 5.1, x ≈ 4 + 5.1 ≈ 9.1 and x ≈ 4 - 5.1 ≈ -1.1

Following these steps ensures a systematic approach to solving any quadratic equation using the quadratic formula, minimizing the chance of errors and maximizing understanding.

Method 2: Completing the Square

Completing the square is another powerful technique for solving quadratic equations. It transforms the equation into a perfect square trinomial, making it easier to isolate the variable. Let's apply this method to −x2+8x+10=0-x^2 + 8x + 10 = 0. First, we want the coefficient of the x² term to be 1. Multiply the entire equation by -1: x² - 8x - 10 = 0. Next, move the constant term to the right side of the equation: x² - 8x = 10. Now, we need to add a value to both sides of the equation to make the left side a perfect square trinomial. To find this value, take half of the coefficient of the x term (which is -8), square it ((-8/2)² = (-4)² = 16), and add it to both sides: x² - 8x + 16 = 10 + 16. This simplifies to x² - 8x + 16 = 26. Now, the left side is a perfect square trinomial and can be factored as (x - 4)² = 26. Take the square root of both sides: x - 4 = ±√26. Finally, isolate x by adding 4 to both sides: x = 4 ± √26. So, we have the solutions x = 4 + √26 and x = 4 - √26, which are the same solutions we found using the quadratic formula. Completing the square is particularly useful because it provides a deeper understanding of the structure of quadratic equations and how they relate to perfect square trinomials. It also forms the basis for deriving the quadratic formula itself. While it may seem more complex than directly applying the quadratic formula, completing the square is a valuable skill for more advanced algebraic manipulations and problem-solving. The process of completing the square involves transforming a quadratic expression into a form that contains a squared term and a constant, enabling the equation to be easily solved by taking square roots. This technique not only solves quadratic equations but also aids in understanding the properties and behavior of quadratic functions.

Step-by-Step Solution by Completing the Square

Let’s break down the process of solving the equation −x2+8x+10=0-x^2 + 8x + 10 = 0 by completing the square. This method involves transforming the quadratic equation into a perfect square trinomial, which simplifies the process of finding the solutions.

  1. Multiply by -1 to make the leading coefficient positive:

    • −x2+8x+10=0-x^2 + 8x + 10 = 0 becomes x2−8x−10=0x^2 - 8x - 10 = 0

  2. Move the constant term to the right side of the equation:

    • x2−8x=10x^2 - 8x = 10

  3. Calculate the value to complete the square:

    • Take half of the coefficient of the x term, which is -8. Half of -8 is -4. Square -4: (−4)2=16(-4)^2 = 16. This is the value we need to add to both sides.

  4. Add the calculated value to both sides of the equation:

    • x2−8x+16=10+16x^2 - 8x + 16 = 10 + 16

  5. Simplify both sides of the equation:

    • x2−8x+16=26x^2 - 8x + 16 = 26

  6. Factor the left side as a perfect square trinomial:

    • (x−4)2=26(x - 4)^2 = 26

  7. Take the square root of both sides:

    • x−4=±√26x - 4 = ±√26

  8. Solve for x:

    • x=4±√26x = 4 ± √26

  9. State the two solutions:

    • x=4+√26x = 4 + √26 and x=4−√26x = 4 - √26

  10. Approximate the solutions (if needed):

    • Since √26 ≈ 5.1, x≈4+5.1≈9.1x ≈ 4 + 5.1 ≈ 9.1 and x≈4−5.1≈−1.1x ≈ 4 - 5.1 ≈ -1.1

Following these steps carefully will lead you to the solutions of the quadratic equation using the completing the square method. This approach not only solves the equation but also enhances understanding of algebraic manipulations.

Conclusion

In summary, we've explored two effective methods for solving the quadratic equation −x2+8x+10=0-x^2 + 8x + 10 = 0: the quadratic formula and completing the square. Both methods lead to the same solutions, x = 4 + √26 and x = 4 - √26. The quadratic formula is a direct, algorithmic approach that works for any quadratic equation. Completing the square provides a deeper understanding of the equation's structure and is valuable for more advanced algebraic manipulations. Understanding and mastering these techniques will equip you to confidently solve a wide range of quadratic equations. Remember, practice makes perfect, so keep honing your skills! For further reading and more advanced topics, visit Khan Academy.