Solving Quadratic Equations Find Solutions For 9x^2 + 10 = 5

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In the realm of mathematics, quadratic equations hold a prominent position, serving as a cornerstone for various applications across diverse fields. Mastering the art of solving these equations is crucial for students, engineers, scientists, and anyone seeking to unravel the intricacies of mathematical problem-solving. This comprehensive guide delves into the world of quadratic equations, equipping you with the knowledge and techniques to confidently tackle any quadratic equation that comes your way. Let's embark on this mathematical journey together!

Understanding Quadratic Equations

At its core, 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:

ax2+bx+c=0ax^2 + bx + c = 0

where 'a', 'b', and 'c' are constants, with 'a' not equal to 0. The coefficients 'a', 'b', and 'c' play a crucial role in determining the nature and solutions of the quadratic equation. The solutions to a quadratic equation are also known as its roots or zeros, representing the values of 'x' that satisfy the equation. These roots correspond to the points where the parabola represented by the quadratic equation intersects the x-axis.

Methods for Solving Quadratic Equations

Over the years, mathematicians have developed a variety of methods for solving quadratic equations, each with its unique approach and applicability. Let's explore some of the most commonly used methods:

1. Factoring

Factoring is a technique that involves expressing the quadratic equation as a product of two linear factors. This method is particularly effective when the quadratic equation can be easily factored. The process involves identifying two numbers that add up to the coefficient 'b' and multiply to the product of 'a' and 'c'. Once the factors are identified, the quadratic equation can be rewritten as a product of two linear factors, and the solutions can be obtained by setting each factor equal to zero and solving for 'x'.

For instance, consider the quadratic equation:

x2+5x+6=0x^2 + 5x + 6 = 0

We need to find two numbers that add up to 5 and multiply to 6. These numbers are 2 and 3. Therefore, we can factor the equation as:

(x+2)(x+3)=0(x + 2)(x + 3) = 0

Setting each factor equal to zero, we get:

x+2=0x + 2 = 0 or x+3=0x + 3 = 0

Solving for 'x', we find the solutions:

x=−2x = -2 or x=−3x = -3

2. Completing the Square

Completing the square is a versatile method that can be used to solve any quadratic equation, regardless of whether it can be factored easily. This technique involves manipulating the quadratic equation to create a perfect square trinomial on one side of the equation. A perfect square trinomial is a trinomial that can be expressed as the square of a binomial.

The steps involved in completing the square are as follows:

  1. Divide the entire equation by the coefficient 'a' (if 'a' is not equal to 1).
  2. Move the constant term 'c' to the right side of the equation.
  3. Take half of the coefficient 'b', square it, and add it to both sides of the equation. This step creates the perfect square trinomial on the left side.
  4. Rewrite the left side as the square of a binomial.
  5. Take the square root of both sides of the equation.
  6. Solve for 'x'.

Let's illustrate this method with an example. Consider the quadratic equation:

2x2+8x−10=02x^2 + 8x - 10 = 0

  1. Divide by 2: x2+4x−5=0x^2 + 4x - 5 = 0
  2. Move the constant term: x2+4x=5x^2 + 4x = 5
  3. Take half of 4 (which is 2), square it (which is 4), and add it to both sides: x2+4x+4=5+4x^2 + 4x + 4 = 5 + 4
  4. Rewrite as a square: (x+2)2=9(x + 2)^2 = 9
  5. Take the square root: x+2=±3x + 2 = \pm 3
  6. Solve for 'x': x=−2±3x = -2 \pm 3, which gives us x=1x = 1 or x=−5x = -5

3. Quadratic Formula

The quadratic formula is a powerful tool that provides a direct solution for any quadratic equation. This formula is derived from the method of completing the square and is universally applicable. The quadratic formula is expressed as:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

To use the quadratic formula, simply identify the coefficients 'a', 'b', and 'c' from the quadratic equation and substitute them into the formula. The formula will then provide the two possible solutions for 'x'. The expression under the square root, b2−4acb^2 - 4ac, is known as the discriminant. The discriminant provides valuable information about the nature of the solutions:

  • If b2−4ac>0b^2 - 4ac > 0, the equation has two distinct real solutions.
  • If b2−4ac=0b^2 - 4ac = 0, the equation has one real solution (a repeated root).
  • If b2−4ac<0b^2 - 4ac < 0, the equation has two complex solutions.

Let's apply the quadratic formula to solve the equation:

3x2−5x+2=03x^2 - 5x + 2 = 0

Here, a=3a = 3, b=−5b = -5, and c=2c = 2. Substituting these values into the quadratic formula, we get:

x=−(−5)±(−5)2−4(3)(2)2(3)x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(3)(2)}}{2(3)}

x=5±25−246x = \frac{5 \pm \sqrt{25 - 24}}{6}

x=5±16x = \frac{5 \pm 1}{6}

This gives us two solutions:

x=5+16=1x = \frac{5 + 1}{6} = 1 and x=5−16=23x = \frac{5 - 1}{6} = \frac{2}{3}

4. Graphical Method

While not always the most precise method, the graphical method provides a visual representation of the solutions to a quadratic equation. This method involves plotting the quadratic equation as a parabola on a coordinate plane. The points where the parabola intersects the x-axis represent the real solutions of the equation. If the parabola does not intersect the x-axis, the equation has no real solutions (it has complex solutions).

To use the graphical method, you can either plot the graph manually by calculating several points or use graphing software or calculators. The graphical method is particularly useful for visualizing the nature of the solutions and understanding the relationship between the quadratic equation and its graph.

Solving the Given Quadratic Equation

Now, let's tackle the specific quadratic equation presented in the question:

9x2+10=59x^2 + 10 = 5

First, we need to rewrite the equation in the standard form ax2+bx+c=0ax^2 + bx + c = 0:

9x2+10−5=09x^2 + 10 - 5 = 0

9x2+5=09x^2 + 5 = 0

Here, we have a=9a = 9, b=0b = 0, and c=5c = 5. Since 'b' is 0, we can solve this equation by isolating x2x^2:

9x2=−59x^2 = -5

x2=−59x^2 = -\frac{5}{9}

Now, take the square root of both sides:

x=±−59x = \pm \sqrt{-\frac{5}{9}}

Since we have a negative value under the square root, the solutions will be complex:

x=±−53x = \pm \frac{\sqrt{-5}}{3}

x=±5i3x = \pm \frac{\sqrt{5}i}{3}

Thus, the solutions are:

x=−5i3x = -\frac{\sqrt{5}i}{3} and x=5i3x = \frac{\sqrt{5}i}{3}

Therefore, the correct answer is C. x=−5i3,5i3x=-\frac{5 i}{3}, \frac{5 i}{3}.

Real-World Applications of Quadratic Equations

Quadratic equations are not just abstract mathematical concepts; they have numerous real-world applications across various disciplines. Here are a few examples:

  • Physics: Quadratic equations are used to model projectile motion, calculate the trajectory of objects thrown into the air, and determine the range and maximum height of projectiles.
  • Engineering: Engineers use quadratic equations in structural design, bridge construction, and optimization problems. For example, they can be used to calculate the optimal dimensions of a bridge arch or the load-bearing capacity of a beam.
  • Economics: Economists use quadratic equations to model cost functions, revenue functions, and profit maximization problems. They can help businesses determine the optimal production levels and pricing strategies.
  • Computer Graphics: Quadratic equations are used extensively in computer graphics to create curves, surfaces, and animations. They are fundamental to rendering realistic images and creating smooth transitions in animations.
  • Finance: Quadratic equations can be used to model investment growth, calculate loan payments, and determine the break-even point for investments.

Tips and Tricks for Solving Quadratic Equations

To enhance your problem-solving skills and tackle quadratic equations with greater ease, consider these helpful tips and tricks:

  • Master the different methods: Familiarize yourself with factoring, completing the square, the quadratic formula, and the graphical method. Each method has its strengths and weaknesses, and choosing the appropriate method can save you time and effort.
  • Practice regularly: Solving quadratic equations requires practice. The more you practice, the more comfortable you will become with the different techniques and the better you will be at identifying the most efficient method for a given problem.
  • Check your solutions: After solving a quadratic equation, always check your solutions by substituting them back into the original equation. This will help you identify any errors and ensure that your solutions are correct.
  • Use the discriminant: The discriminant can provide valuable information about the nature of the solutions. Use it to determine whether the equation has two distinct real solutions, one real solution, or two complex solutions.
  • Visualize the graph: If possible, visualize the graph of the quadratic equation. This can help you understand the relationship between the equation and its solutions and provide a visual check for your answers.

Conclusion

Solving quadratic equations is a fundamental skill in mathematics with wide-ranging applications. By mastering the various methods, understanding the nature of solutions, and practicing regularly, you can confidently tackle any quadratic equation that comes your way. Remember to choose the most appropriate method for each problem, check your solutions, and visualize the graph whenever possible. With dedication and practice, you will become a proficient quadratic equation solver!