Solving Equations A Step By Step Guide To Kent's Method

Hey guys! Let's dive into this interesting math problem where Kent uses a clever trick to solve an equation. We're going to break it down step by step, so you'll not only understand the solution but also the reasoning behind it. This is super useful for tackling similar problems in the future.

The Initial Equation

So, we start with Kent's equation:

k + 12/k = 8

This looks a bit tricky, right? We've got a fraction in there, which can make things messy. But don't worry, Kent has a plan! He decides to multiply both sides of the equation by an expression to get rid of that fraction. This is a common technique in algebra – eliminating fractions to simplify the equation. By multiplying, Kent aims to transform this equation into something more manageable, likely a quadratic equation that we can easily solve.

Why Multiply?

The key here is to think about what's causing the issue. It's the 12/k term, so we want to get rid of the k in the denominator. Multiplying the entire equation by k will do the trick. When we multiply k by 12/k, the k in the numerator and denominator will cancel out, leaving us with just 12. This is a classic algebraic manipulation – using multiplication to clear fractions.

Multiplying Both Sides

So, Kent multiplies both sides of the equation by k. This gives us:

k * (k + 12/k) = 8 * k

Remember, whatever we do to one side of the equation, we must do to the other side to keep things balanced. It's like a scale – if you add weight to one side, you need to add the same weight to the other side to keep it level. This principle ensures that the equation remains valid and that we're not changing the solution.

Distributing and Rearranging

Now, let's distribute the k on the left side of the equation. We get:

k^2 + 12 = 8k

See how the fraction is gone? We've successfully cleared the denominator! Now the equation looks much simpler. But we're not quite there yet. The next step is to move all the terms to one side of the equal sign. This is a crucial step in solving quadratic equations because it allows us to set the equation equal to zero, which is the standard form for solving quadratics.

Moving Terms

Kent wants to move all the terms to one side, so he subtracts 8k from both sides of the equation. This gives us:

k^2 - 8k + 12 = 0

Now we have a quadratic equation in standard form: ax^2 + bx + c = 0, where a = 1, b = -8, and c = 12. This is the form we need to solve for k. This step is essential because many techniques for solving quadratic equations, such as factoring or using the quadratic formula, rely on having the equation in this standard form.

The Resulting Equation

So, the equation Kent must solve is:

k^2 - 8k + 12 = 0

This corresponds to option A. $k^2-8 k-12=0$

Solving the Quadratic Equation

Now that we have the quadratic equation, we can solve it using various methods. Let's explore a couple of common approaches: factoring and the quadratic formula. These methods are fundamental in algebra, and mastering them will significantly boost your problem-solving skills.

Factoring

Factoring involves breaking down the quadratic expression into two binomial expressions. We need to find two numbers that multiply to give the constant term (12) and add up to the coefficient of the linear term (-8). In this case, the numbers are -2 and -6 because (-2) * (-6) = 12 and (-2) + (-6) = -8. So, we can rewrite the equation as:

(k - 2)(k - 6) = 0

Now, for the product of two factors to be zero, at least one of them must be zero. This gives us two possible solutions:

k - 2 = 0  or  k - 6 = 0

Solving these equations, we get:

k = 2  or  k = 6

So, the solutions to the quadratic equation are k = 2 and k = 6. Factoring is a powerful technique, but it's most effective when the quadratic equation has integer solutions. If the solutions are not integers, the quadratic formula is a more reliable method.

The Quadratic Formula

The quadratic formula is a general method for solving quadratic equations of the form ax^2 + bx + c = 0. The formula is:

k = (-b ± √(b^2 - 4ac)) / (2a)

In our case, a = 1, b = -8, and c = 12. Plugging these values into the formula, we get:

k = (8 ± √((-8)^2 - 4 * 1 * 12)) / (2 * 1)

Simplifying, we have:

k = (8 ± √(64 - 48)) / 2
k = (8 ± √16) / 2
k = (8 ± 4) / 2

This gives us two solutions:

k = (8 + 4) / 2 = 6
k = (8 - 4) / 2 = 2

As you can see, we get the same solutions (k = 2 and k = 6) as we did with factoring. The quadratic formula is a versatile tool that works for any quadratic equation, regardless of whether the solutions are integers or not.

Key Takeaways

Alright, guys, let's recap what we've learned in this problem. This is super important for solidifying your understanding and applying these techniques to future problems.

Clearing Fractions

The first crucial step was clearing the fraction by multiplying both sides of the equation by k. This is a common strategy in algebra to simplify equations involving fractions. By eliminating the denominator, we transformed the equation into a more manageable form. Remember, always multiply every term on both sides of the equation to maintain balance.

Rearranging to Standard Form

Next, we rearranged the equation to the standard quadratic form: ax^2 + bx + c = 0. This form is essential for both factoring and using the quadratic formula. Moving all terms to one side and setting the equation to zero allows us to apply these standard methods effectively. This step ensures that we can use established techniques for solving quadratic equations.

Factoring and the Quadratic Formula

We explored two methods for solving the quadratic equation: factoring and the quadratic formula. Factoring is a quick method when the solutions are integers, but the quadratic formula is a more general approach that works for any quadratic equation. Knowing both methods gives you flexibility in solving problems. The quadratic formula is particularly useful when factoring is not straightforward.

Checking Your Solutions

Finally, it's always a good idea to check your solutions by plugging them back into the original equation. This ensures that your solutions are correct and that you haven't made any mistakes along the way. Substituting the values back into the original equation confirms that they satisfy the equation, giving you confidence in your answer.

Practice Makes Perfect

Solving equations like this might seem challenging at first, but with practice, you'll become more comfortable and confident. The key is to understand the underlying principles and apply them systematically. Keep practicing, and you'll master these techniques in no time! Remember, math is like a muscle – the more you use it, the stronger it gets.

So, next time you encounter a similar equation, remember Kent's trick: clear the fractions, rearrange to standard form, and then solve using factoring or the quadratic formula. You've got this!