Solving Quadratic Equations 2x² - 8x + 6 Find The Zeros
Hey everyone! Let's dive into solving a quadratic equation today. We've got the expression 2x² - 8x + 6, and our mission is to find which value of 'x' makes this whole thing equal to zero. Basically, we need to find the roots or zeros of this quadratic equation. The options given are A) 1, B) 3, C) 4, and D) 5. So, let's break it down step by step!
Understanding Quadratic Equations
Before we jump into solving this specific problem, let's get a quick refresher on what quadratic equations are all about. A quadratic equation is an equation of the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. These equations pop up all over the place in math and real-world applications, from figuring out the trajectory of a ball to designing bridges. The solutions to a quadratic equation are also known as its roots or zeros, which are the values of 'x' that make the equation true.
Why are Quadratic Equations Important?
Quadratic equations aren't just abstract math problems; they have real-world applications that make them super important. Here are a few areas where they show up:
- Physics: When you're calculating the path of a projectile, like a ball thrown in the air, you're dealing with quadratic equations. They help describe motion under gravity.
- Engineering: Engineers use quadratic equations to design structures, ensuring they're stable and can withstand forces. Think about bridges and buildings – quadratic equations play a role in their design.
- Computer Graphics: In the world of computer graphics, quadratic equations are used to create curves and surfaces. This is crucial for rendering realistic images and animations.
- Economics: Quadratic equations can model cost, revenue, and profit in business scenarios. They help in making informed decisions about pricing and production.
Key Components of a Quadratic Equation
Let's break down the anatomy of a quadratic equation:
- ax²: This is the quadratic term. The coefficient 'a' determines the shape of the parabola (the U-shaped curve you get when you graph a quadratic equation). If 'a' is positive, the parabola opens upwards, and if 'a' is negative, it opens downwards.
- bx: This is the linear term. The coefficient 'b' affects the parabola's position in the coordinate plane.
- c: This is the constant term. It determines the y-intercept of the parabola, which is the point where the parabola intersects the y-axis.
Understanding these components helps in visualizing and solving quadratic equations effectively. Now that we've got a handle on the basics, let's move on to the methods we can use to find those elusive roots!
Methods to Solve Quadratic Equations
Okay, so how do we actually find the values of 'x' that make our quadratic expression equal to zero? There are several methods we can use, and each has its own strengths. Let's explore a few common ones:
1. Factoring
Factoring is like the superhero method when it works. It's fast and elegant, but it's not always applicable. The idea is to rewrite the quadratic expression as a product of two binomials. If we can factor the expression, we can easily find the roots.
How it works:
- Look for two numbers that multiply to give 'c' (the constant term) and add up to 'b' (the coefficient of the 'x' term).
- Rewrite the quadratic expression using these numbers.
- Set each factor equal to zero and solve for 'x'.
For example, let's say we have the equation x² - 5x + 6 = 0. We need two numbers that multiply to 6 and add to -5. Those numbers are -2 and -3. So, we can rewrite the equation as (x - 2)(x - 3) = 0. Setting each factor to zero gives us x = 2 and x = 3.
2. Quadratic Formula
If factoring is the superhero method, the quadratic formula is the Swiss Army knife of quadratic equation solving. It works every single time, no matter how messy the equation is. It might look a bit intimidating at first, but it's a reliable tool to have in your math arsenal.
The Formula:
The quadratic formula is given by:
Where 'a', 'b', and 'c' are the coefficients from our quadratic equation ax² + bx + c = 0.
How to use it:
- Identify 'a', 'b', and 'c' from your equation.
- Plug these values into the quadratic formula.
- Simplify the expression to find the two possible values for 'x'.
3. Completing the Square
Completing the square is a method that's a bit more involved, but it's a great way to understand the structure of quadratic equations. It's also useful in other areas of math, like calculus.
How it works:
- Rewrite the equation in the form (x + p)² + q = 0.
- Solve for 'x'.
This method involves manipulating the equation to create a perfect square trinomial, which can then be easily solved. It's a bit like transforming the equation into a more manageable form.
4. Graphing
Sometimes, the easiest way to find the roots of a quadratic equation is to visualize it. Graphing the equation allows us to see where the parabola intersects the x-axis, which gives us the roots.
How to do it:
- Graph the quadratic equation y = ax² + bx + c.
- Identify the points where the parabola crosses the x-axis. These are the roots of the equation.
Graphing can be done by hand or using graphing software. It's a great way to get a visual understanding of the solutions.
Now that we've explored these methods, let's apply one to our specific problem and find out which value of 'x' zeros the expression 2x² - 8x + 6!
Solving 2x² - 8x + 6 = 0
Alright, let's get our hands dirty and solve the equation 2x² - 8x + 6 = 0. We have a few options here, but I think factoring might be the quickest route. However, to make things even simpler, let's first see if we can simplify the equation a bit.
Step 1: Simplify the Equation
Notice that all the coefficients (2, -8, and 6) are divisible by 2. So, let's divide the entire equation by 2. This will give us a simpler equation to work with:
2x² - 8x + 6 = 0
Divide by 2:
x² - 4x + 3 = 0
Step 2: Factoring the Simplified Equation
Now, we've got a simpler equation: x² - 4x + 3 = 0. Let's try factoring this. We need two numbers that multiply to 3 and add up to -4. Those numbers are -1 and -3.
So, we can rewrite the equation as:
(x - 1)(x - 3) = 0
Step 3: Find the Roots
Now, we set each factor equal to zero and solve for 'x':
x - 1 = 0 or x - 3 = 0
Solving these gives us:
x = 1 or x = 3
Step 4: Check the Options
Looking back at our options, we see that both 1 (Option A) and 3 (Option B) are solutions. So, the values of 'x' that zero the expression are 1 and 3.
Justification
To justify our answer, let's plug these values back into the original equation and see if they make it equal to zero.
For x = 1:
2(1)² - 8(1) + 6 = 2 - 8 + 6 = 0
For x = 3:
2(3)² - 8(3) + 6 = 2(9) - 24 + 6 = 18 - 24 + 6 = 0
Both values work! So, we've successfully found the zeros of the quadratic expression.
Conclusion
So, guys, the values of 'x' that make the expression 2x² - 8x + 6 equal to zero are 1 and 3. We found this by simplifying the equation, factoring it, and then solving for 'x'. We also justified our answer by plugging the values back into the original equation and confirming that they work. Remember, quadratic equations are all around us, and knowing how to solve them is a valuable skill. Keep practicing, and you'll become a quadratic equation-solving pro in no time!
