What Is The Only Solution Of 2x2 8x X2 16

What is the only solution of 2x2 8x x2 16?
A step‑by‑step guide to solving the quadratic equation that yields a single (double) root

When you first glance at the string “2x2 8x x2 16” it can look like a jumble of numbers and letters. In algebra, however, such a pattern usually hides a simple quadratic equation once the missing operation signs are inserted. The most common interpretation that leads to a single solution is

[ 2x^{2}+8x = x^{2}-16 . ]

Below we will unpack this equation, solve it rigorously, and explain why it has exactly one answer. The discussion is written for high‑school students, adult learners, or anyone who wants to see the logical flow from a confusing expression to a clear result.

Introduction: Why the Equation Matters

Quadratic equations appear everywhere—from physics formulas for projectile motion to finance models for profit maximization. Recognizing when a quadratic collapses to a double root (i.e., only one distinct solution) is a valuable skill because it tells us the parabola just touches the x‑axis instead of crossing it.

The phrase “what is the only solution of 2x2 8x x2 16” is essentially asking: If we interpret the given symbols as a quadratic equation, what value of x satisfies it, and why is there no other value?

Our main keyword—what is the only solution of 2x2 8x x2 16—will appear naturally throughout the article to help search engines understand the focus while keeping the text readable for humans.

Understanding the Equation

1. Inserting the Missing Operators

The original string lacks plus, minus, and equals signs. By looking at the structure we can guess the intended form:

“2x2” → (2x^{2})
“8x” → (+8x) (or (-8x); we will test both)
“x2” → (x^{2}) - “16” → constant term

A placement that yields a single solution is:

[ 2x^{2}+8x = x^{2}-16 . ]

If we moved all terms to the left‑hand side we obtain a standard quadratic:

[ 2x^{2}+8x - x^{2}+16 = 0 \quad\Longrightarrow\quad x^{2}+8x+16 = 0 . ]

2. Recognizing the Perfect Square

The left‑hand side (x^{2}+8x+16) is a perfect‑square trinomial because:

[ (x+4)^{2}=x^{2}+2\cdot4\cdot x+4^{2}=x^{2}+8x+16 . ]

Thus the equation simplifies to:

[ (x+4)^{2}=0 . ]

A squared quantity equals zero only when the quantity itself is zero, giving us the only possible value for (x).

Step‑by‑Step Solution

Below is a detailed walk‑through that you can follow with pen and paper or a calculator.

Step	Action	Reasoning
1	Write the equation with clear operators: (2x^{2}+8x = x^{2}-16).	This is the most plausible interpretation that leads to a unique root.
2	Bring every term to one side: subtract (x^{2}) and add 16 to both sides.	We want the standard form (ax^{2}+bx+c=0).
3	Simplify: (2x^{2}-x^{2}+8x+16 = 0) → (x^{2}+8x+16 = 0).	Combine like terms.
4	Identify the quadratic coefficients: (a=1), (b=8), (c=16).	Needed for the discriminant or factoring.
5	Compute the discriminant (\Delta = b^{2}-4ac).	(\Delta = 8^{2}-4\cdot1\cdot16 = 64-64 = 0).
6	Because (\Delta = 0), the quadratic has one real double root.	A zero discriminant

Because the squared term ((x+4)^{2}) can only be zero when its base is zero, we set

[ x+4 = 0 ;\Longrightarrow; x = -4 . ]

Substituting (-4) back into the original arrangement confirms the equality:

[ 2(-4)^{2}+8(-4) = 2\cdot16-32 = 0,\qquad (-4)^{2}-16 = 16-16 = 0, ]

so both sides match. No other real number can satisfy ((x+4)^{2}=0); any deviation from (-4) makes the square positive, breaking the equality. Hence the equation possesses a single, double root at (x=-4).

Why Other Placements Fail to Yield a Unique Solution

If we had chosen a different arrangement of the symbols—such as (2x^{2}-8x = x^{2}+16) or (2x^{2}+8x = -(x^{2}+16))—the resulting quadratic would have a non‑zero discriminant ((\Delta>0) or (\Delta<0)), producing either two distinct real roots or a pair of complex conjugates. Only the specific grouping that leads to (x^{2}+8x+16=0) yields (\Delta=0), the hallmark of a double root.

Connecting Back to the Key Phrase The question “what is the only solution of 2x2 8x x2 16” invites us to reconstruct the missing operators, recognize the perfect‑square pattern, and apply the discriminant test. By doing so we uncover that the sole value satisfying the equation is (-4), and we understand why no alternative value can work.

Conclusion

Interpreting the cryptic string (2x2;8x;x2;16) as the quadratic equation (2x^{2}+8x = x^{2}-16) simplifies to ((x+4)^{2}=0). Because a square equals zero only when its base is zero, the equation admits exactly one real solution:

[ \boxed{x=-4}. ]

This double root tells us that the corresponding parabola merely touches the x‑axis at ((-4,0)) without crossing it—a key insight for applications ranging from projectile motion to profit‑maximization models. Thus, the answer to “what is the only solution of 2x2 8x x2 16” is uniquely (-4).