Solving Quadratic Equations By Completing The Square A Step By Step Guide

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Hey guys! Today, we're diving deep into the fascinating world of quadratic equations and tackling them using a method called "completing the square." It might sound intimidating, but trust me, it's a super useful technique to have in your math toolkit. We'll break it down step-by-step, using the example equation $7x - 9 = 7x^2 - 49x$ to guide us. So, buckle up, and let's get started!

Understanding Quadratic Equations

Before we jump into solving, let's quickly recap what quadratic equations are. Quadratic equations are polynomial equations of the second degree, meaning the highest power of the variable (usually x) is 2. The standard form of a quadratic equation is $ax^2 + bx + c = 0$, where a, b, and c are constants, and a is not equal to 0. These equations pop up everywhere in math and science, from calculating the trajectory of a ball to designing bridges. Mastering how to solve them is key for any aspiring mathematician or engineer.

Why Completing the Square?

You might be wondering, why bother with completing the square when we have other methods like factoring or the quadratic formula? Well, completing the square is more than just a problem-solving technique; it's a fundamental concept that helps us understand the structure of quadratic equations. It allows us to rewrite the equation in a form that reveals the vertex of the parabola (the graph of a quadratic equation) and makes it easier to solve in certain situations, especially when factoring isn't straightforward. Plus, it's the backbone behind the derivation of the quadratic formula itself! So, understanding completing the square gives you a deeper insight into the nature of quadratic equations.

Step-by-Step Solution: Completing the Square

Now, let's get our hands dirty and solve the equation $7x - 9 = 7x^2 - 49x$ by completing the square. We'll go through each step in detail so you can follow along easily.

Step 1: Rearrange the Equation

Our first goal is to get the equation into a form where all the x terms are on one side and the constant term is on the other. This means we need to move the $7x$ term from the left side to the right side. We can do this by subtracting $7x$ from both sides of the equation:

7xβˆ’9βˆ’7x=7x2βˆ’49xβˆ’7x7x - 9 - 7x = 7x^2 - 49x - 7x

This simplifies to:

βˆ’9=7x2βˆ’56x-9 = 7x^2 - 56x

Now we have the constant term (-9) isolated on the left side, and the quadratic and linear terms (those with x squared and x, respectively) on the right side. This sets us up nicely for the next step.

Step 2: Factor out the Leading Coefficient

This step is crucial when the coefficient of the $x^2$ term (which is 7 in our case) is not 1. We need to factor out this coefficient from the terms on the right side of the equation. This will make it easier to complete the square inside the parentheses.

So, we factor out 7 from $7x^2 - 56x$:

βˆ’9=7(x2βˆ’8x)-9 = 7(x^2 - 8x)

Notice how we factored out the 7, leaving us with $x^2 - 8x$ inside the parentheses. This is exactly what we want! We're now one step closer to creating a perfect square trinomial.

Step 3: Completing the Square

This is the heart of the method! To complete the square, we need to add a constant term inside the parentheses that will turn the expression $x^2 - 8x$ into a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored as $(x + a)^2$ or $(x - a)^2$.

To find this constant, we take half of the coefficient of the x term (which is -8), square it, and add it inside the parentheses. Half of -8 is -4, and (-4) squared is 16. So, we add 16 inside the parentheses:

-9 + ext{_____} = 7(x^2 - 8x + ext{_____})

-9 + ext{_____} = 7(x^2 - 8x + 16)

But here's the catch: we can't just add 16 to one side of the equation without changing its value. Since we're adding 16 inside the parentheses, which are being multiplied by 7, we're actually adding 7 * 16 = 112 to the right side of the equation. To keep the equation balanced, we must also add 112 to the left side:

βˆ’9+112=7(x2βˆ’8x+16)-9 + 112 = 7(x^2 - 8x + 16)

Now we have a balanced equation with a perfect square trinomial inside the parentheses.

Step 4: Factor the Perfect Square Trinomial

The expression inside the parentheses, $x^2 - 8x + 16$, is now a perfect square trinomial. This means it can be factored into the form $(x - a)^2$. In our case, it factors as:

x2βˆ’8x+16=(xβˆ’4)2x^2 - 8x + 16 = (x - 4)^2

So, our equation becomes:

103=7(xβˆ’4)2103 = 7(x - 4)^2

We've successfully transformed the equation into a much simpler form! Notice how the perfect square trinomial has allowed us to express the x terms as a squared expression.

Step 5: Isolate the Squared Term

To continue solving for x, we need to isolate the squared term, $(x - 4)^2$. We can do this by dividing both sides of the equation by 7:

rac{103}{7} = (x - 4)^2

Now we have the squared term all by itself on one side of the equation.

Step 6: Take the Square Root of Both Sides

To get rid of the square, we take the square root of both sides of the equation. Remember that when we take the square root, we need to consider both the positive and negative roots:

Β±1037=xβˆ’4\pm \sqrt{\frac{103}{7}} = x - 4

This gives us two possible solutions for x.

Step 7: Solve for x

Finally, we isolate x by adding 4 to both sides of the equation:

x=4Β±1037x = 4 \pm \sqrt{\frac{103}{7}}

This gives us our two solutions:

x = 4 + \sqrt{\frac{103}{7}}$ and $x = 4 - \sqrt{\frac{103}{7}}

These are the roots of the quadratic equation $7x - 9 = 7x^2 - 49x$. We've successfully solved it by completing the square!

Key Takeaways

Completing the square might seem like a lot of steps, but it's a powerful technique that provides a deep understanding of quadratic equations. Here are the key takeaways to remember:

  • Rearrange the equation to get the constant term on one side and the x terms on the other.
  • Factor out the leading coefficient if it's not 1.
  • Complete the square by adding $(b/2)^2$ inside the parentheses, where b is the coefficient of the x term.
  • Remember to balance the equation by adding the same value to both sides.
  • Factor the perfect square trinomial.
  • Isolate the squared term.
  • Take the square root of both sides (remembering both positive and negative roots).
  • Solve for x.

Practice Makes Perfect

The best way to master completing the square is to practice! Try solving different quadratic equations using this method. You'll get more comfortable with the steps, and you'll start to see the patterns and nuances involved. Don't be afraid to make mistakes – that's how we learn! And remember, completing the square is not just about finding the solutions; it's about understanding the structure and properties of quadratic equations. So, keep practicing, and you'll become a quadratic equation pro in no time!

Conclusion

So, there you have it! We've walked through the entire process of solving a quadratic equation by completing the square. I hope this step-by-step guide has been helpful and has demystified this important technique. Remember, math is like building a house – each concept builds upon the previous one. By mastering completing the square, you're strengthening the foundation for your future mathematical adventures. Keep exploring, keep learning, and most importantly, keep having fun with math!