Solving Quadratic Equations Step-by-Step Guide

Hey everyone! Let's dive into the fascinating world of quadratic equations. These equations, characterized by the presence of a squared variable (like x²), pop up in various fields, from physics to engineering to even everyday problem-solving. In this article, we'll tackle two specific quadratic equations, breaking down the steps and concepts involved. So, grab your pencils, and let's get started!

Understanding Quadratic Equations

Before we jump into solving, let's quickly recap what quadratic equations are. A quadratic equation is essentially a polynomial equation of the second degree. This means the highest power of the variable (usually 'x') is 2. The standard form of a quadratic equation is:

ax² + bx + c = 0

Where 'a', 'b', and 'c' are coefficients (numbers), and 'a' cannot be zero (otherwise, it wouldn't be a quadratic equation!). The solutions to a quadratic equation are also known as its roots or zeros. These are the values of 'x' that make the equation true.

There are several methods to solve quadratic equations, including factoring, completing the square, and the quadratic formula. We'll be using these methods as we work through our examples. Keep in mind, a quadratic equation can have two distinct real solutions, one real solution (a repeated root), or two complex solutions.

Now that we have a basic understanding, let's move on to our first equation.

Solving 3x = 0.5x²

Our first equation is: 3x = 0.5x². To solve this, we need to rearrange it into the standard form of a quadratic equation (ax² + bx + c = 0). This involves moving all the terms to one side of the equation.

Step 1: Rearrange the Equation

Subtract 3x from both sides of the equation:

0. 5x² - 3x = 0

Now, we have our equation in the standard form, where a = 0.5, b = -3, and c = 0. Notice that 'c' is zero in this case, which will simplify our solution process.

Step 2: Factoring

In this case, factoring is the easiest method to use. We can see that 'x' is a common factor in both terms. Let's factor out 'x':

x(0.5x - 3) = 0

Step 3: Apply the Zero Product Property

The Zero Product Property states that if the product of two factors is zero, then at least one of the factors must be zero. In our case, the factors are 'x' and '(0.5x - 3)'. So, we can set each factor equal to zero and solve for 'x':

• x = 0
• 0. 5x - 3 = 0

Step 4: Solve for x

The first solution is already straightforward: x = 0.

For the second solution, let's solve the equation 0.5x - 3 = 0:

1. Add 3 to both sides: 0.5x = 3
2. Divide both sides by 0.5: x = 3 / 0.5 = 6

Step 5: State the Solutions

Therefore, the solutions to the equation 3x = 0.5x² are x = 0 and x = 6. These are the two values of 'x' that make the original equation true. We can verify this by substituting these values back into the original equation and seeing if the equation holds.

Verification

Let's check our solutions:

• For x = 0: 3(0) = 0.5(0)², which simplifies to 0 = 0. This is true.
• For x = 6: 3(6) = 0.5(6)², which simplifies to 18 = 0.5(36) = 18. This is also true.

So, our solutions are correct! We've successfully solved our first quadratic equation using factoring. Now, let's move on to the second equation, which might require a different approach.

Solving 0 = 5x² - 2x + 6

Our second equation is: 0 = 5x² - 2x + 6. This equation is already in the standard form (ax² + bx + c = 0), where a = 5, b = -2, and c = 6. In this case, factoring might not be the most straightforward method, so let's explore using the quadratic formula.

The Quadratic Formula

The quadratic formula is a powerful tool that can solve any quadratic equation, regardless of whether it can be easily factored. The 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

As we mentioned earlier, for our equation 0 = 5x² - 2x + 6, we have:

• a = 5
• b = -2
• c = 6

Step 2: Substitute into the Quadratic Formula

Now, let's substitute these values into the quadratic formula:

x = (-(-2) ± √((-2)² - 4 * 5 * 6)) / (2 * 5)

Step 3: Simplify

Let's simplify the expression step by step:

x = (2 ± √(4 - 120)) / 10 x = (2 ± √(-116)) / 10

Step 4: Handle the Negative Square Root

Notice that we have a negative number under the square root (√(-116)). This means the solutions will be complex numbers. Recall that the square root of -1 is denoted by 'i' (the imaginary unit). We can rewrite √(-116) as:

√(-116) = √(116 * -1) = √(116) * √(-1) = √(116) * i

We can further simplify √(116) by finding its prime factorization. 116 = 2 * 2 * 29, so √(116) = √(2² * 29) = 2√(29).

Therefore, √(-116) = 2√(29)i

Step 5: Write the Solutions

Now we can rewrite our solutions as:

x = (2 ± 2√(29)i) / 10

We can simplify this further by dividing both the real and imaginary parts by 2:

x = (1 ± √(29)i) / 5

Step 6: State the Solutions

So, the solutions to the equation 0 = 5x² - 2x + 6 are:

• x = (1 + √(29)i) / 5
• x = (1 - √(29)i) / 5

These are two complex solutions. This means there are no real numbers that satisfy this quadratic equation. The solutions involve the imaginary unit 'i'.

Key Takeaways

In this article, we successfully solved two quadratic equations using different methods. Here are the key takeaways:

• Quadratic equations are equations of the form ax² + bx + c = 0.
• They can have two real solutions, one real solution (a repeated root), or two complex solutions.
• Factoring is a useful method when the equation can be easily factored.
• The quadratic formula (x = (-b ± √(b² - 4ac)) / (2a)) can solve any quadratic equation.
• Complex solutions arise when the discriminant (b² - 4ac) is negative.

Practice Makes Perfect

Solving quadratic equations is a fundamental skill in algebra. The more you practice, the more comfortable you'll become with different methods and types of equations. Try solving more quadratic equations on your own, and don't hesitate to review the concepts and steps we've discussed here.

Keep practicing, guys, and you'll become quadratic equation masters in no time! Remember to always double-check your solutions and understand the underlying concepts. Good luck, and happy solving!