What Is the Solution to 2x² + 8x + x² = 16?
You're looking at the equation 2x² + 8x + x² = 16, and maybe you're wondering what on earth x is supposed to be. Because of that, here's the thing — this is a quadratic equation, and once you know how to approach it, the solution is straightforward. Let me walk you through it.
Understanding the Equation
First, let's clean this up. Think about it: you have 2x² + 8x + x² = 16. That said, those two x² terms? They combine. 2x² + x² = 3x².
3x² + 8x = 16
Now we need to get everything on one side so it equals zero — that's the standard form for solving quadratics:
3x² + 8x - 16 = 0
This is the equation we're actually solving. We subtracted 16 from both sides to set everything to zero. See how it works? That's the key move That alone is useful..
Why Quadratics Matter
Here's why this matters. Which means quadratic equations show up everywhere — physics problems involving projectile motion, business calculations for profit maximization, even video game programming. Understanding how to solve them isn't just some abstract math skill; it's a tool that pops up in real situations.
The catch? Worth adding: they don't realize that with a few simple steps, you can find exactly what x equals. Because of that, most people get stuck when they see x² and x in the same equation. And once you know the process, you can handle any quadratic that comes your way.
How to Solve It
There are a few ways to solve 3x² + 8x - 16 = 0. Let me show you the two most practical approaches.
Method 1: Factoring
First, check if the equation can be factored. We need two numbers that multiply to (3)(-16) = -48 and add up to 8 Most people skip this — try not to. Worth knowing..
Think: what multiplies to -48 and adds to 8?
12 and -4 work. 12 × (-4) = -48, and 12 + (-4) = 8 Still holds up..
Now we can split the middle term:
3x² + 12x - 4x - 16 = 0
Group them:
(3x² + 12x) + (-4x - 16) = 0
Factor each group:
3x(x + 4) - 4(x + 4) = 0
Now factor out the common (x + 4):
(3x - 4)(x + 4) = 0
Set each factor to zero:
3x - 4 = 0 → x = 4/3
x + 4 = 0 → x = -4
So we get x = 4/3 or x = -4 Easy to understand, harder to ignore..
Method 2: The Quadratic Formula
Sometimes factoring is tricky, and that's when the quadratic formula saves you. For any equation ax² + bx + c = 0, the formula is:
x = (-b ± √(b² - 4ac)) / 2a
For our equation, a = 3, b = 8, and c = -16. Plugging in:
x = (-8 ± √(8² - 4(3)(-16))) / 2(3) x = (-8 ± √(64 + 192)) / 6 x = (-8 ± √256) / 6 x = (-8 ± 16) / 6
This gives us the same two solutions:
x = (-8 + 16) / 6 = 8/6 = 4/3
x = (-8 - 16) / 6 = -24/6 = -4
Same answers. Different path That's the part that actually makes a difference..
Wait — Why Two Solutions?
You asked about "the only solution," but here's what most people miss: quadratic equations typically have two solutions. That's just how they work. Both x = 4/3 and x = -4 are valid answers to the original equation.
Now, depending on the context, one might make more sense than the other. Which means if you're solving a real-world problem, negative values sometimes don't make sense (you can't have -4 apples, for instance). But mathematically? Both are correct And that's really what it comes down to..
Common Mistakes to Avoid
Here's where people go wrong:
1. Forgetting to set the equation to zero. You can't factor or use the formula if you still have "= 16" sitting there. Always move everything to one side first.
2. Not combining like terms. Seeing 2x² + x² and treating them as separate is a sure way to get stuck. Combine them to get 3x² The details matter here. That's the whole idea..
3. Making sign errors. When you move 16 to the left side, it becomes -16. Simple, but easy to mess up when you're moving fast.
4. Stopping after finding one solution. Factoring gives you factors like (3x - 4)(x + 4) = 0. You need to set each factor equal to zero. Missing one means missing a solution Which is the point..
Practical Tips
- Always simplify first. Combine like terms, get everything on one side. Clean equations are easier to solve.
- Check your work by plugging your answers back into the original equation. For x = 4/3: 2(16/9) + 8(4/3) + (16/9) = 32/9 + 32/3 + 16/9 = 32/9 + 96/9 + 16/9 = 144/9 = 16. It works.
- If factoring feels hard, just use the quadratic formula. It never fails.
- Write down every step. Quadratics have a lot of moving parts, and skipping steps is where errors creep in.
FAQ
Does the equation 2x² + 8x + x² = 16 have only one solution?
No. So it has two solutions: x = 4/3 and x = -4. Quadratic equations typically have two solutions.
What's the fastest way to solve this?
Use the quadratic formula if you're unsure about factoring. It's reliable and works every time Took long enough..
Can I verify my answers?
Absolutely. Also, plug each solution back into the original equation 2x² + 8x + x² = 16. Both should give you 16.
Why do quadratics have two solutions?
Because x appears squared, which means both positive and negative values can satisfy the equation. That's just the math.
What if I only got one answer?
Check if you set both factors to zero. With (3x - 4)(x + 4) = 0, you need to solve 3x - 4 = 0 AND x + 4 = 0 Simple, but easy to overlook..
The Bottom Line
The equation 2x² + 8x + x² = 16 simplifies to 3x² + 8x - 16 = 0, and its solutions are x = 4/3 and x = -4. Both are valid. The key is simplifying first, choosing a reliable method (factoring or the quadratic formula), and making sure you solve from each factor. Once you see the pattern, quadratics stop being intimidating — they're just a few steps, and you've got this.
Short version: it depends. Long version — keep reading.
