Find The Quadratic Equation With Given Solutions

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Hey guys! Ever stumbled upon a math problem that looks like it’s speaking another language? Well, today we’re diving into one of those! We need to figure out which quadratic equation has the solutions $\frac{5 \pm 2 \sqrt{7}}{3}$. Sounds like fun, right? Let's break it down and make it super easy to understand. We'll explore how to reverse-engineer the solutions to find the correct equation. We're not just solving for x today; we're going to decode the math behind it! Get ready to put on your math hats, and let’s get started!

Understanding the Problem

Okay, first things first. We’re given two solutions: $\frac{5 + 2 \sqrt{7}}{3}$ and $\frac{5 - 2 \sqrt{7}}{3}$. These solutions come from a quadratic equation, which generally looks like $ax^2 + bx + c = 0$. Our mission, should we choose to accept it (and we do!), is to find the correct values for $a$, $b$, and $c$ that match the provided options. Remember, the quadratic formula is our friend here: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. We’re essentially working backward from the solutions to reconstruct the original equation. This involves some algebraic gymnastics, but don’t worry, we’ll take it step by step. We need to manipulate these solutions to find relationships that help us pinpoint the right equation. Think of it as being a math detective, piecing together clues to solve the mystery of the missing equation. We’ll be using properties of roots and coefficients to make this process smoother and more intuitive.

Method 1: Using the Sum and Product of Roots

Sum and Product of Roots: A classic and super handy technique! If we have a quadratic equation $ax^2 + bx + c = 0$, the sum of the roots (let's call them $x_1$ and $x_2$) is $-\frac{b}{a}$, and the product of the roots is $\frac{c}{a}$.

Let's apply this to our solutions:

• $x_1 = \frac{5 + 2 \sqrt{7}}{3}$
• $x_2 = \frac{5 - 2 \sqrt{7}}{3}$

Sum of the roots:

$x_1 + x_2 = \frac{5 + 2 \sqrt{7}}{3} + \frac{5 - 2 \sqrt{7}}{3} = \frac{5 + 2 \sqrt{7} + 5 - 2 \sqrt{7}}{3} = \frac{10}{3}$

So, $-\frac{b}{a} = \frac{10}{3}$. This tells us that $b = -10k$ and $a = 3k$ for some constant $k$. This narrows down our choices to equations C and D, which both have $a = 3$ and $b = -10$.

Product of the roots:

$x_1 * x_2 = \frac{5 + 2 \sqrt{7}}{3} * \frac{5 - 2 \sqrt{7}}{3} = \frac{(5 + 2 \sqrt{7})(5 - 2 \sqrt{7})}{9} = \frac{25 - (4 * 7)}{9} = \frac{25 - 28}{9} = \frac{-3}{9} = -\frac{1}{3}$

So, $\frac{c}{a} = -\frac{1}{3}$. Since we know $a = 3$, then $c = -1$. Thus, our equation looks like $3x^2 - 10x - 1 = 0$.

Therefore, option D is the correct answer.

Method 2: Constructing the Quadratic Equation

Constructing the Equation: Another way to solve this is by constructing the quadratic equation directly from its roots. If $x_1$ and $x_2$ are the roots of a quadratic equation, then the equation can be written as $(x - x_1)(x - x_2) = 0$.

Using our solutions:

• $x_1 = \frac{5 + 2 \sqrt{7}}{3}$
• $x_2 = \frac{5 - 2 \sqrt{7}}{3}$

We have:

$\left(x - \frac{5 + 2 \sqrt{7}}{3}\right)\left(x - \frac{5 - 2 \sqrt{7}}{3}\right) = 0$

To get rid of the fractions, we can multiply both sides by 3:

$\left(3x - (5 + 2 \sqrt{7})\right)\left(3x - (5 - 2 \sqrt{7})\right) = 0$

Now, expand the expression:

$(3x - 5 - 2 \sqrt{7})(3x - 5 + 2 \sqrt{7}) = 0$

This looks like the difference of squares, $(A - B)(A + B) = A^2 - B^2$, where $A = (3x - 5)$ and $B = 2 \sqrt{7}$.

So, we have:

$(3x - 5)^2 - (2 \sqrt{7})^2 = 0$

Expanding further:

$(9x^2 - 30x + 25) - (4 * 7) = 0$

$9x^2 - 30x + 25 - 28 = 0$

$9x^2 - 30x - 3 = 0$

Now, divide the entire equation by 3 to simplify:

$3x^2 - 10x - 1 = 0$

Again, we find that option D is the correct answer.

Step-by-Step Verification

Let's double-check: Just to be absolutely sure, let's use the quadratic formula on option D, $3x^2 - 10x - 1 = 0$, and see if we get our original solutions.

The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

In our case, $a = 3$, $b = -10$, and $c = -1$.

Plugging these values in:

$x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(3)(-1)}}{2(3)}$

$x = \frac{10 \pm \sqrt{100 + 12}}{6}$

$x = \frac{10 \pm \sqrt{112}}{6}$

Since $\sqrt{112} = \sqrt{16 * 7} = 4 \sqrt{7}$, we have:

$x = \frac{10 \pm 4 \sqrt{7}}{6}$

Divide both the numerator and the denominator by 2:

$x = \frac{5 \pm 2 \sqrt{7}}{3}$

Yes! This matches our original solutions. So, we can confidently say that option D is indeed the correct answer.

Why Other Options Are Incorrect

To reinforce our understanding, let's briefly examine why the other options are incorrect:

• Option A: $3x^2 - 5x + 7 = 0$: The sum and product of roots would be different from what we calculated. Specifically, the sum of the roots would be $\frac{5}{3}$, which doesn't match our required $\frac{10}{3}$.
• Option B: $3x^2 - 5x - 1 = 0$: Similar to option A, the sum of the roots would be $\frac{5}{3}$, and the product of the roots would be $-\frac{1}{3}$. While the product matches, the sum does not, making this option incorrect.
• Option C: $3x^2 - 10x + 6 = 0$: Here, the sum of the roots would be $\frac{10}{3}$, which matches our required sum. However, the product of the roots would be $\frac{6}{3} = 2$, which does not match our required $-\frac{1}{3}$.

By systematically analyzing each option, we can clearly see why option D is the only one that aligns with both the sum and product of the given roots.

Conclusion

Alright, mathletes! We’ve successfully navigated this quadratic equation problem using a couple of cool methods: the sum and product of roots, and constructing the equation directly from its solutions. We also verified our answer to make sure we’re 100% correct. Remember, when tackling these problems, always look for ways to simplify and use the properties of equations to your advantage. Keep practicing, and you’ll become a math whiz in no time! Until next time, keep those numbers crunching! Option D is the final answer.