Solve: Quadratic Equation With Given Solutions

Hey Plastik Magazine readers! Ever wondered how to reverse-engineer a quadratic equation when you're given the solutions? It's like being a math detective, and today, we're cracking the case. We're going to take you step-by-step through finding the quadratic equation that has the solutions ${x = \frac{5 \pm 2 \sqrt{7}}{3}}$. Let's dive in!

Understanding the Problem

So, what's the big idea here? We have two solutions, and we need to find the quadratic equation that spits them out. Remember, a quadratic equation generally looks like this: ${ax^2 + bx + c = 0}$. Our mission, should we choose to accept it, is to find the values of ${a}$, ${b}$, and ${c}$. We have been provided with the solutions in the form of ${x = \frac{5 \pm 2 \sqrt{7}}{3}}$. This form screams quadratic formula, which is ${x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}$. The strategy is to manipulate our given solutions to match the quadratic formula and then extract the coefficients.

Breaking Down the Solution

First, let's rewrite the given solutions to resemble the quadratic formula more closely. We have:

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

To make it look more like the quadratic formula, we want to isolate the ${\pm}$ part:

${x = \frac{5 \pm \sqrt{4 \cdot 7}}{3} = \frac{5 \pm \sqrt{28}}{3}}$

Now, we want the denominator to be in the form of ${2a}$, so we compare it with the general quadratic formula:

${x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}$

From this, we can infer some relationships. We can see that ${2a}$ should correspond to 3. However, this might not be immediately obvious, and we might need to manipulate the equation further to match the form perfectly. What’s crucial is to relate the terms inside the square root and the constant term outside the square root.

Detailed Comparison with the Quadratic Formula

Let's dig deeper into the comparison. We have ${x = \frac{5 \pm \sqrt{28}}{3}}$. We want to express this in the form of the quadratic formula ${x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}$. By comparing the two, we can start making educated guesses about the values of ${a}$, ${b}$, and ${c}$.

If we assume ${a = 3/2}$ then ${2a = 3}$, which matches our denominator. But since we prefer integer coefficients, let's aim to manipulate the equation to get integer values for ${a}$, ${b}$, and ${c}$.

Let's try to rewrite our solution to have a '2' in the denominator. To do this, we can multiply both the numerator and denominator by 2/2:

${x = \frac{2}{2} \cdot \frac{5 \pm \sqrt{28}}{3} = \frac{10 \pm 2\sqrt{28}}{6} = \frac{10 \pm \sqrt{4 \cdot 28}}{6} = \frac{10 \pm \sqrt{112}}{6}}$

Now we have ${x = \frac{10 \pm \sqrt{112}}{6}}$. Comparing this with ${x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}$, we can deduce that ${2a = 6}$, so ${a = 3}$. This is great because we now have an integer value for ${a}$.

Next, we can see that ${-b = 10 \implies b = -10}$

Now we have ${a = 3}$ and ${b = -10}$. We can use the discriminant (the part under the square root) to find ${c}$. We have:

${b^2 - 4ac = 112}$

Plugging in our values for ${a}$ and ${b}$:

${(-10)^2 - 4(3)c = 112}$

${100 - 12c = 112}$

${-12c = 12}$

${c = -1}$

So we have ${a = 3}$, ${b = -10}$, and ${c = -1}$. Plugging these into our quadratic equation ${ax^2 + bx + c = 0}$, we get:

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

Therefore, the correct equation is ${3x^2 - 10x - 1 = 0}$.

Checking the Answer

To be absolutely sure, let's use the quadratic formula on our derived equation ${3x^2 - 10x - 1 = 0}$ and see if we get back our original solutions. The quadratic formula is:

${x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}$

Plugging in ${a = 3}$, ${b = -10}$, and ${c = -1}$:

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

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

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

${x = \frac{10 \pm \sqrt{16 \cdot 7}}{6}}$

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

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

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

Voila! We got back our original solutions. This confirms that our equation ${3x^2 - 10x - 1 = 0}$ is indeed correct.

The Final Answer

The quadratic equation that has the solutions ${x = \frac{5 \pm 2 \sqrt{7}}{3}}$ is:

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

So the correct option is D.

Key Takeaways

• Reverse Engineering: Given solutions, you can work backward to find the quadratic equation.
• Quadratic Formula: Understanding and recognizing the quadratic formula is crucial.
• Matching Coefficients: Carefully compare the given solutions with the quadratic formula to deduce the coefficients.
• Verification: Always verify your equation by plugging the coefficients back into the quadratic formula.

Why This Matters

Understanding quadratic equations isn't just about acing your math exams; it's a fundamental skill that pops up in various fields. From physics to engineering to even financial modeling, quadratic equations help describe and solve real-world problems. Mastering the ability to reverse-engineer these equations gives you a deeper understanding of how they work and their applications.

Real-World Applications

• Physics: Projectile motion is described by quadratic equations. Knowing the solutions (where the projectile lands) can help you determine the initial velocity and angle.
• Engineering: Designing structures and systems often involves solving quadratic equations to ensure stability and efficiency.
• Finance: Modeling investments and calculating returns can utilize quadratic equations to optimize strategies.

Tips for Success

• Practice, Practice, Practice: The more you work with quadratic equations, the easier it becomes to recognize patterns and apply the correct methods.
• Understand the Formula: Memorizing the quadratic formula is important, but understanding where it comes from is even better.
• Check Your Work: Always verify your solutions by plugging them back into the original equation.
• Stay Organized: Keep your work neat and organized to avoid mistakes.

So, there you have it, mathletes! Cracking quadratic equations, one step at a time. Keep practicing, stay curious, and remember: math can be fun, especially when you're rocking it with Plastik Magazine! Keep your eyes peeled for more math adventures, and until next time, keep those calculators handy!