Solve 2x² = -10x + 12: Find The Quadratic Solutions
Hey guys! Ever stumbled upon a quadratic equation that looks like a jumbled mess of numbers and variables? Don't sweat it! Quadratic equations might seem intimidating at first, but with a few simple steps, you can crack them like a pro. In this guide, we'll tackle the equation 2x² = -10x + 12 head-on. We'll break down the process, explore different methods, and find the solutions (also known as roots) together. So, buckle up, grab your thinking caps, and let's dive into the world of quadratic equations!
Understanding Quadratic Equations
Before we jump into solving, let's get a grip on what a quadratic equation actually is. At its core, a quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (usually 'x') is 2. The general form of a quadratic equation is ax² + bx + c = 0, where 'a', 'b', and 'c' are constants (numbers) and 'a' cannot be zero (otherwise, it wouldn't be a quadratic equation anymore!).
Think of it like this: a quadratic equation represents a curve called a parabola when graphed. The solutions (roots) of the equation are the points where the parabola intersects the x-axis. These points are also known as the x-intercepts or zeros of the equation. Finding these solutions is the name of the game!
Why are quadratic equations important, you ask? Well, they pop up in all sorts of places in the real world, from physics and engineering to economics and finance. They can help us model projectile motion, optimize designs, and even predict market trends. So, mastering quadratic equations is a valuable skill to have in your mathematical toolkit.
In our specific equation, 2x² = -10x + 12, we can see that it fits the quadratic equation mold. Our mission is to rearrange it into the standard form (ax² + bx + c = 0) and then find the values of 'x' that make the equation true. Let's get to it!
Method 1: Solving by Factoring
Factoring is a classic method for solving quadratic equations, and it's often the quickest route when it works. The key idea behind factoring is to rewrite the quadratic expression as a product of two linear expressions (expressions of the form (x + something) or (x - something)).
Step 1: Rearrange the Equation
Our first task is to get the equation into the standard form, ax² + bx + c = 0. To do this, we need to move all the terms to one side of the equation, leaving zero on the other side. In our case, we have 2x² = -10x + 12. Let's add 10x and subtract 12 from both sides:
2x² + 10x - 12 = 0
Now we have our equation in the standard form. Great job!
Step 2: Simplify by Factoring out the Greatest Common Factor (GCF)
Before we jump into factoring the quadratic expression, let's see if we can simplify things by factoring out the greatest common factor (GCF) from all the terms. Looking at our equation, 2x² + 10x - 12 = 0, we can see that all the coefficients (the numbers in front of the variables) are divisible by 2. So, let's factor out a 2:
2(x² + 5x - 6) = 0
This makes our equation a bit easier to work with. Now we can focus on factoring the expression inside the parentheses: x² + 5x - 6.
Step 3: Factor the Quadratic Expression
Now comes the heart of the factoring method. We need to find two numbers that:
- Multiply to the constant term (-6 in our case)
- Add up to the coefficient of the x term (5 in our case)
Think of it like a puzzle. We need to find the right pieces that fit together. After a bit of thought (or trial and error), you might realize that the numbers 6 and -1 fit the bill:
- 6 * -1 = -6
- 6 + (-1) = 5
So, we can rewrite the quadratic expression as a product of two binomials (expressions with two terms):
x² + 5x - 6 = (x + 6)(x - 1)
Step 4: Set Each Factor to Zero and Solve
Now we have our equation in factored form:
2(x + 6)(x - 1) = 0
The key principle here is the zero-product property: if the product of several factors is zero, then at least one of the factors must be zero. This means we can set each factor equal to zero and solve for x:
- x + 6 = 0 => x = -6
- x - 1 = 0 => x = 1
And there you have it! We've found two solutions to our quadratic equation: x = -6 and x = 1. These are the roots of the equation.
Method 2: Using the Quadratic Formula
The quadratic formula is a powerful tool that can solve any quadratic equation, no matter how messy it looks. It's like a universal key that unlocks the solutions. While factoring is great when it works, the quadratic formula is always a reliable backup.
The quadratic formula is given by:
x = (-b ± √(b² - 4ac)) / 2a
Where 'a', 'b', and 'c' are the coefficients from the standard form of the quadratic equation (ax² + bx + c = 0).
Step 1: Identify a, b, and c
First, we need to identify the values of 'a', 'b', and 'c' in our equation. Remember, our equation in standard form is 2x² + 10x - 12 = 0. So:
- a = 2 (the coefficient of x²)
- b = 10 (the coefficient of x)
- c = -12 (the constant term)
Step 2: Plug the Values into the Quadratic Formula
Now comes the fun part: plugging the values into the formula. Let's carefully substitute the values of 'a', 'b', and 'c' into the quadratic formula:
x = (-10 ± √(10² - 4 * 2 * -12)) / (2 * 2)
Step 3: Simplify the Expression
Next, we need to simplify the expression. Let's break it down step by step:
- x = (-10 ± √(100 + 96)) / 4
- x = (-10 ± √196) / 4
- x = (-10 ± 14) / 4
Step 4: Calculate the Two Possible Solutions
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