What Are The Solutions Of X 2 6x 6 10

Solving the Quadratic Equation: x² + 6x + 6 = 10

This article delves into the comprehensive solution of the quadratic equation x² + 6x + 6 = 10. We'll explore various methods to find the values of 'x' that satisfy this equation, including factoring, completing the square, using the quadratic formula, and graphically interpreting the solution. Understanding these different approaches is crucial for mastering quadratic equations and their applications in various fields.

Understanding Quadratic Equations

Before diving into the solution, let's briefly review the fundamental characteristics of quadratic equations. A quadratic equation is a second-degree polynomial equation of the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. Our given equation, x² + 6x + 6 = 10, needs to be rearranged into this standard form before we can apply different solution methods.

Rearranging the Equation

The first step is to rearrange the given equation, x² + 6x + 6 = 10, into the standard form ax² + bx + c = 0. To do this, we subtract 10 from both sides of the equation:

x² + 6x + 6 - 10 = 0

This simplifies to:

x² + 6x - 4 = 0

Now that our equation is in standard form (with a=1, b=6, and c=-4), we can proceed with different solution methods.

Method 1: Factoring

Factoring is a method used to find the roots of a quadratic equation by expressing it as a product of two linear factors. Unfortunately, this particular quadratic equation, x² + 6x - 4 = 0, cannot be easily factored using integers. There are no two numbers that add up to 6 and multiply to -4. Therefore, we need to explore other methods.

Method 2: Completing the Square

Completing the square is a powerful technique to solve quadratic equations. It involves manipulating the equation to create a perfect square trinomial, which can then be easily factored. Let's apply this method to our equation:

1. Move the constant term: Move the constant term (-4) to the right-hand side of the equation:

   x² + 6x = 4

2. Find the value to complete the square: Take half of the coefficient of 'x' (which is 6), square it ((6/2)² = 9), and add it to both sides of the equation:

   x² + 6x + 9 = 4 + 9

3. Factor the perfect square trinomial: The left side of the equation is now a perfect square trinomial:

   (x + 3)² = 13

4. Solve for x: Take the square root of both sides:

   x + 3 = ±√13

5. Isolate x: Subtract 3 from both sides:

   x = -3 ± √13

Therefore, the solutions are x = -3 + √13 and x = -3 - √13. These are the exact solutions. Approximate values can be obtained using a calculator: x ≈ 0.6056 and x ≈ -6.6056.

Method 3: Quadratic Formula

The quadratic formula is a general formula that provides the solutions for any quadratic equation in the form ax² + bx + c = 0. The formula is:

x = [-b ± √(b² - 4ac)] / 2a

Substituting the values from our equation (a=1, b=6, c=-4) into the quadratic formula, we get:

x = [-6 ± √(6² - 4 * 1 * -4)] / (2 * 1)

x = [-6 ± √(36 + 16)] / 2

x = [-6 ± √52] / 2

x = [-6 ± 2√13] / 2

Simplifying, we get:

x = -3 ± √13

This confirms the solutions we obtained using the completing the square method: x = -3 + √13 and x = -3 - √13.

Method 4: Graphical Interpretation

The solutions of a quadratic equation can also be found graphically. The x-intercepts (points where the graph intersects the x-axis) represent the solutions to the equation. Plotting the graph of y = x² + 6x - 4 will show two x-intercepts, corresponding to the values of x we calculated using the algebraic methods. While this method provides a visual representation of the solutions, it's less precise than the algebraic methods, particularly for equations with non-integer solutions.

Understanding the Discriminant

The discriminant (b² - 4ac) within the quadratic formula provides valuable information about the nature of the solutions:

• If b² - 4ac > 0: The equation has two distinct real roots (as in our case).
• If b² - 4ac = 0: The equation has one real root (a repeated root).
• If b² - 4ac < 0: The equation has two complex roots (roots involving imaginary numbers).

In our equation, the discriminant is 52 (36 + 16), which is greater than 0, indicating two distinct real roots.

Applications of Quadratic Equations

Quadratic equations have widespread applications in various fields, including:

• Physics: Calculating projectile motion, determining the trajectory of objects under gravity.
• Engineering: Designing structures, analyzing stress and strain on materials.
• Economics: Modeling supply and demand, optimizing production.
• Computer Graphics: Creating curves and shapes.

Understanding how to solve quadratic equations is therefore essential for tackling problems in these and many other areas.

Conclusion

We've explored four different methods—factoring (though not applicable in this specific case), completing the square, the quadratic formula, and graphical interpretation—to solve the quadratic equation x² + 6x + 6 = 10. The solutions, x = -3 + √13 and x = -3 - √13, were obtained using the most efficient methods. The concept of the discriminant was also discussed to highlight its significance in understanding the nature of solutions. Mastering these techniques is crucial for successfully tackling more complex mathematical problems and real-world applications. Remember to always check your solutions by substituting them back into the original equation to verify their accuracy. Further exploration into more complex quadratic equations and their applications will solidify your understanding of this fundamental mathematical concept. Practice consistently to improve your speed and accuracy in solving such problems.