What are the solutions of the equation x⁴ - 5x² - 14 = 0? Use factoring to solve.

Solution:

Step1:

Given: Equation is x⁴ - 5x² - 14 = 0 --- (1)

As the equation is of the fourth degree

Substitute x⁴ = u² or x² =u

So that the equation changes to a quadratic equation.

Step2:

After substitution, the equation changes to

u² - 5u - 14 = 0 --- (2)

Which is quadratic equation u

Equation (2) can be solved by the factorization method

Step3:

Let us factorize the quadratic equation by splitting the middle term.

The factors of - 14 are - 7 and 2 so that on adding the factors -7 + 2 = - 5, middle term

⇒ u² - 7u + 2u - 14 = 0

Separating the common terms from first two terms and then from next two terms separately

⇒u(u - 7) + 2( u - 7) = 0

⇒ ( u - 7)(u + 2) = 0

⇒ u - 7 = 0 and u + 2 =0

⇒ u = 7 and u = -2

Step-4:

⇒ x² = 7 and x²= -2

⇒ x = ±√7 and x = ±√2i (since √(-1) = ±i)

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What are the solutions of the equation x⁴ - 5x² - 14 = 0 = 0? Use factoring to solve.

Summary:

The values of x satisfying the equation x⁴ - 5x² - 14 = 0 are x = ±√7 and x = ±√2i. The number of roots of an equation depends on the degree of the equation. If the degree of the equation is 4 then the equation has 4 roots.