**What are the solutions of the equation x4 – 5×2 – 14 = 0? Use factoring to solve.**

- x = ±√7 and x = ±√2i
- x=7 and x=5i

### The Correct Answer is

a. x = ±√7 and x = ±√2i

**Explanation of why x = ±√7 and x = ±√2i are the correct answers:**

**Solving x⁴ – 5x² – 14 = 0 using factoring:**

The equation x⁴ – 5x² – 14 = 0 can be factored by using a substitution. Let’s denote y as x². Therefore, the equation becomes y² – 5y – 14 = 0, which is a quadratic equation in y. Factoring it:

(y – 7)(y + 2) = 0

Setting each factor equal to zero:

y – 7 = 0 or y + 2 = 0

Solving for y:

y = 7 or y = -2

Remembering that y = x²:

x² = 7 or x² = -2

Taking the square root of both sides:

x = ±√7 or x = ±√(-2)

However, the square root of a negative number (√(-2)) results in complex numbers, denoted by ‘i’. Thus, x = ±√7 or x = ±√2i.

**x = ±√7:**This solution comes from setting y = 7 (from the factored equation) and taking its square root. This is a valid real number solution.**x = ±√2i:**This solution arises from the fact that y = -2 in the factored equation. Taking the square root of a negative value results in an imaginary number, denoted by ‘i’. Therefore, x = ±√2i is also a valid solution.

**Explanation of why the other provided answers are not correct:**

**x = 7:**This answer does not consider the ± (plus/minus) sign. The correct solutions include both the positive and negative values because the original equation is a fourth-degree polynomial.**x = 5i:**This answer is incorrect because it misses the square root (√) part. The solutions involving imaginary numbers should contain the square root symbol along with ‘i’.

The correct solutions, x = ±√7 and x = ±√2i, encompass both positive and negative values as per the nature of the original equation. The incorrect solutions either neglect the ± sign or fail to express the solutions as square roots when dealing with imaginary numbers.

Understanding how to factor and solve polynomial equations helps in identifying valid solutions and ensures the inclusion of all possible roots based on the nature of the equation.

Solving equations through factoring is a foundational skill in mathematics, enabling us to break down complex equations into simpler components to determine their roots or solutions.

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