Solving $5x^2 + 20x - 7 = 0$ By Completing The Square

Are you wrestling with quadratic equations and the method of completing the square? You're not alone! It's a powerful technique, but it can seem tricky at first. In this article, we'll break down the process step by step, using the example equation $5x^2 + 20x - 7 = 0$. By the end, you'll have a clear understanding of how to solve quadratic equations using this method.

Understanding the Goal: Completing the Square

Before diving into the specifics of our example equation, let’s quickly recap what completing the square actually means. The goal is to manipulate a quadratic equation into a form where one side is a perfect square trinomial. A perfect square trinomial can be factored into the square of a binomial, like $(x + a)^2$. This form makes it much easier to isolate the variable x and find the solutions to the equation.

Why Completing the Square Matters

You might be wondering, “Why bother with completing the square when we have the quadratic formula?” That's a valid question! While the quadratic formula is a reliable workhorse, completing the square offers some unique advantages:

• Conceptual Understanding: Completing the square provides a deeper understanding of the structure of quadratic equations and their solutions. It reveals how the solutions relate to the coefficients of the equation.
• Deriving the Quadratic Formula: Interestingly, the quadratic formula itself is derived by completing the square on the general quadratic equation $ax^2 + bx + c = 0$. So, mastering completing the square gives you insight into the origins of the formula.
• Vertex Form: Completing the square is essential for converting a quadratic equation into vertex form, which directly reveals the vertex (minimum or maximum point) of the parabola represented by the equation. This is incredibly useful in various applications, such as optimization problems.

Now, let's get our hands dirty with our example equation and see how completing the square works in practice.

Step 1: Isolating the Constant Term

The first key step in solving $5x^2 + 20x - 7 = 0$ by completing the square involves isolating the constant term on one side of the equation. We want to get the terms with x on one side and the constant term on the other. To do this, we simply add 7 to both sides of the equation:

$5x^2 + 20x - 7 + 7 = 0 + 7$

This simplifies to:

$5x^2 + 20x = 7$

This might seem like a small step, but it sets the stage for the next crucial manipulation. We've now created space to work with the x terms and transform them into a perfect square trinomial. So, option C. $5(x^2 + 4x) = 7$ is indeed a correct step in the process. Notice how this step also prepares us to deal with the leading coefficient of the $x^2$ term, which we'll tackle in the next step.

Step 2: Factoring Out the Leading Coefficient

The next critical step in completing the square for the equation $5x^2 + 20x = 7$ is to factor out the leading coefficient from the terms containing x. In this case, the leading coefficient is 5. Factoring it out allows us to work with a simpler quadratic expression inside the parentheses.

So, we factor out 5 from the left side of the equation:

$5(x^2 + 4x) = 7$

This step is crucial because the process of completing the square works most directly when the coefficient of the $x^2$ term is 1. By factoring out the 5, we've effectively normalized the quadratic expression inside the parentheses. Now, we can focus on completing the square for the expression $x^2 + 4x$. This involves finding the constant term that, when added to this expression, will create a perfect square trinomial.

Why is this step important?

If we didn't factor out the leading coefficient, we'd have to deal with it throughout the completing the square process, which would make the calculations more complicated. Factoring it out simplifies the process and reduces the chances of making errors. This step aligns perfectly with option C, solidifying its role in the solution.

Step 3: Completing the Square Inside the Parentheses

Now comes the heart of the method: completing the square. We have the equation $5(x^2 + 4x) = 7$. We need to figure out what constant to add inside the parentheses to turn the expression $x^2 + 4x$ into a perfect square trinomial.

Here's the key idea: Take half of the coefficient of the x term (which is 4 in this case), square it, and add that to the expression. Half of 4 is 2, and 2 squared is 4. So, we need to add 4 inside the parentheses.

However, there's a crucial detail to remember! We're not just adding 4 to the left side of the equation. We're adding 4 inside the parentheses, which are being multiplied by 5. So, we're actually adding 5 * 4 = 20 to the left side of the equation. To maintain the balance of the equation, we must also add 20 to the right side.

This gives us:

$5(x^2 + 4x + 4) = 7 + 20$

So, option A. $5(x^2 + 4x + 4) = -7 + 20$ is also a correct step. However, notice the slight discrepancy: the right side should be $7 + 20$, not $-7 + 20$. This highlights the importance of careful attention to detail when working through these steps. While the structure of the left side is correct, the arithmetic on the right side in option A is incorrect.

Perfect Square Trinomial

Now, the expression inside the parentheses, $x^2 + 4x + 4$, is a perfect square trinomial. It can be factored as $(x + 2)^2$. This is the whole point of completing the square – we've created an expression that we can easily rewrite in a more convenient form.

Step 4: Factoring and Simplifying

Now that we've completed the square, let's rewrite the equation and simplify. We have:

$5(x^2 + 4x + 4) = 7 + 20$

First, we factor the perfect square trinomial:

$5(x + 2)^2 = 27$

This is a significant step forward. We've transformed the original quadratic equation into a form where the variable x appears only once, inside the squared term. This makes it much easier to isolate x.

Step 5: Isolating the Squared Term

To continue solving for x, we need to isolate the squared term, $(x + 2)^2$. To do this, we divide both sides of the equation by 5:

rac{5(x + 2)^2}{5} = rac{27}{5}

This simplifies to:

(x + 2)^2 = rac{27}{5}

Now, we're getting closer to isolating x. We have a squared term on one side and a constant on the other. The next logical step is to take the square root of both sides.

Step 6: Taking the Square Root

Taking the square root of both sides of the equation (x + 2)^2 = rac{27}{5} is a crucial step in solving for x. Remember that when we take the square root of both sides, we need to consider both the positive and negative roots. This is because both a positive and a negative number, when squared, will result in a positive number.

So, taking the square root of both sides gives us:

x + 2 = oxed{\pm \sqrt{\frac{27}{5}}}

Therefore, option B. $x + 2 = \pm \sqrt{\frac{27}{5}}$ is a correct step in the process. This equation now represents two separate equations:

• $x + 2 = \sqrt{\frac{27}{5}}$
• $x + 2 = -\sqrt{\frac{27}{5}}$

Each of these equations will lead to a different solution for x.

Step 7: Isolating x

The final step in solving for x is to isolate it by subtracting 2 from both sides of the equation:

$x = -2 \pm \sqrt{\frac{27}{5}}$

These are the two solutions to the quadratic equation $5x^2 + 20x - 7 = 0$. We can further simplify the radical if desired, but this form clearly shows the two roots of the equation.

Conclusion

We've successfully navigated the process of solving the quadratic equation $5x^2 + 20x - 7 = 0$ by completing the square. We identified the crucial steps: isolating the constant term, factoring out the leading coefficient, completing the square, factoring the perfect square trinomial, isolating the squared term, taking the square root, and finally, isolating x.

By understanding each step and the reasoning behind it, you can confidently tackle other quadratic equations using this powerful technique. Remember to practice consistently, and don't be afraid to review the steps when needed. Completing the square might seem challenging initially, but with practice, it will become a valuable tool in your mathematical arsenal.

For more in-depth explanations and examples, you might find it helpful to explore resources like Khan Academy's Quadratic Equations section.