The solutions to \(2x^2 + 8x + 4 = 0\) are \(x = \frac{-8 \pm \sqrt{32}}{4}\).

The solutions to \(2x^2 + 8x + 4 = 0\) are \(x = \frac{-8 \pm \sqrt{32}}{4}\).

December 12, 2024

Question: The solutions to 2x2+8x+4=02x^2 + 8x + 4 = 0 are x=−8±324x = \frac{-8 \pm \sqrt{32}}{4}. Choose the simplified form of these solutions.

A. x=−2±42\displaystyle x = -2 \pm 4\sqrt{2}

B. x=−2±2x = -2 \pm \sqrt{2}

C. x=−8±224\displaystyle x = \frac{-8 \pm 2\sqrt{2}}{4}

D. x=−8±1624\displaystyle x = \frac{-8 \pm 16\sqrt{2}}{4}

Answer: B. x=−2±2x = -2 \pm \sqrt{2}

Explanation:

Step1: Simplify the Square Root

The given solution is:

x=−8±324x = \frac{-8 \pm \sqrt{32}}{4}

First, simplify 32\sqrt{32}:

32=16×2=42\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}

Step2: Substitute the Simplified Root Back into the Equation

Replace 32\sqrt{32} with 424\sqrt{2}:

x=−8±424x = \frac{-8 \pm 4\sqrt{2}}{4}

Step3: Simplify the Fraction

Divide both terms in the numerator by 4:

x=−84±424=−2±2x = \frac{-8}{4} \pm \frac{4\sqrt{2}}{4} = -2 \pm \sqrt{2}

Thus, the simplified form of the solutions is:

x=−2±2x = -2 \pm \sqrt{2}

Extended Knowledge:

Simplifying Square Roots

Simplifying square roots involves breaking down the number under the radical into a product of a perfect square and another number. For example:

32=16×2=16×2=42\sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2}

Types of Roots in Quadratic Solutions

Quadratic equations can yield solutions involving:

  • Simple Roots: Like 2\sqrt{2}, which are straightforward and easy to interpret.
  • Complex Roots: In cases where the discriminant is negative, leading to imaginary numbers.

Importance of Simplifying Solutions

Simplifying solutions:

  • Enhances Clarity: Makes the solutions easier to understand and communicate.
  • Reduces Computational Complexity: Simplified forms are often more manageable in further calculations.
  • Standardizes Format: Ensures consistency, especially when comparing or combining solutions.

Practical Application

Simplified solutions are crucial in various fields:

  • Engineering: For precise measurements and calculations.
  • Physics: When deriving formulas and interpreting experimental data.
  • Finance: In modeling and predicting financial trends based on quadratic relationships.

Conclusion

Option B correctly represents the simplified form of the solutions to the quadratic equation 2x2+8x+4=02x^2 + 8x + 4 = 0. Simplifying the square root and the resulting fraction leads to a clear and concise expression:

x=−2±2x = -2 \pm \sqrt{2}

This process underscores the importance of simplifying mathematical expressions to facilitate better understanding and application.