Find the value of k, for which one root of the quadratic equation kx² - 14x + 8 = 0 is six times the other.

Zeros and roots are the values that satisfy a given equation. The term zeros are used for an algebraic expression whereas roots are used for a given equation.

Answer: The value of k for which one root of the quadratic equation kx² - 14x + 8 = 0 is six times the other is k = 3.

Let's look into the solution step by step.

Explanation:

Given: A quadratic equation, kx² - 14x + 8 = 0

Let the two zeros of the equation be α and β.

According to the given question, if one of the roots is α the other root will be 6α.

Thus, β = 6α

Hence, the two zeros are α and 6α.

We know that for a given quadratic equation ax² + bx + c = 0

The sum of the zeros is expressed as,

α + β = - b / a

The product of the zeros is expressed as,

αβ = c / a

For the given quadratic equation kx² - 14x + 8 = 0,

a = k, b = -14, c = 8

The sum of the zeros is:

α + 6α = 14 / k [Since the two zeros are α and 6α]

⇒ 7α  = 14 / k

⇒ α = 2 / k --------------- (1)

The product of the zeros is:

⇒ α × 6α  = 8 / k [Since the two zeros are α and 6α] 

⇒ 6α ² = 8 / k

⇒ 6 (2 / k)² = 8 / k  [From (1)]

⇒ 6 × (4 / k) = 8

⇒ k = 24 / 8

⇒ k = 3

We can also use Cuemath's online roots calculator to find the zeros of a polynomial.

Thus, the value of k is equal to 8 for the given quadratic equation kx² - 14x + 8 = 0.