Question

The smallest value of \(k\), for which both the roots of the equation $$ x^{2}-8 k x+16\left(k^{2}-k+1\right)=0 $$ are real, distinct and have values at least 4 , is

Short answer

The smallest value of k for which the roots are real, distinct, and at least 4 is just greater than 1.

Step-by-step solution

Determine the Conditions for Real and Distinct Roots

For the roots of a quadratic equation of the form ax^2 + bx + c = 0 to be real and distinct, the discriminant must be positive, where the discriminant is given by D = b^2 - 4ac. In this case, a = 1, b = -8k, and c = 16(k^2 - k + 1). Therefore, D = (-8k)^2 - 4(1)(16(k^2 - k + 1)).

Calculate the Discriminant

Calculate the discriminant D using the coefficients from the given equation: D = (-8k)^2 - 4(1)(16(k^2 - k + 1)) = 64k^2 - 64(k^2 - k + 1) = 64k^2 - 64k^2 + 64k - 64. For the roots to be real and distinct, D must be greater than 0.

Apply the Condition for Real and Distinct Roots

Set D > 0 and solve for k: 64k^2 - 64k^2 + 64k - 64 > 0 reduces to 64k - 64 > 0. Dividing by 64 gives k - 1 > 0, which further simplifies to k > 1.

Apply the Condition for Roots to be at Least 4

If the roots are at least 4, they must be greater than or equal to 4, so by Vieta's formulas, the sum of the roots, which is -b/a, must be greater than or equal to 8. With a = 1 and b = -8k, -b/a >= 8 leads to 8k >= 8, so k >= 1.

Combine the Conditions to Determine the Smallest Value of k

Combining the conditions from Step 3 and Step 4, we find that k must satisfy both k > 1 and k >= 1. Therefore, the smallest value of k that satisfies both conditions is k > 1.

Key concepts

Discriminant in Quadratic Equations

Understanding the discriminant is crucial when studying quadratic equations. The discriminant gives us critical information about the nature of the roots without actually solving the equation. For any quadratic equation of the form ax^2 + bx + c = 0, the discriminant is expressed as D = b^2 - 4ac.

The value of the discriminant tells us:

• If D > 0, the equation has two distinct real roots.
• If D = 0, the equation has exactly one real root, also called a repeated or double root.
• If D < 0, the equation has no real roots, indicating two complex conjugate roots.

In the provided exercise, by analyzing the discriminant, we ensure the roots are not only real but also distinct, laying the groundwork for further conditions to satisfy the given problem. This direct relationship between the discriminant and the nature of roots simplifies the process of defining the parameter k.

Vieta's Formulas

Vieta's formulas are a set of equations that relate the coefficients of a polynomial to sums and products of its roots. For a quadratic equation ax^2 + bx + c = 0, Vieta's formulas state:

• The sum of the roots (-b/a) is equal to the opposite of the coefficient of x, divided by the coefficient of x^2.
• The product of the roots (c/a) equals the constant term, divided by the coefficient of x^2.

Using these formulas, we don't need to solve the equation to find the roots; instead, we can instantly infer information about the roots based on the equation's coefficients. In our case, Vieta’s formulas aid in determining that the value of k has to be at least 1 to ensure the sum of the roots is greater than or equal to 8, which aligns with the additional condition that each root must be at least 4.

Conditions for Real and Distinct Roots

The conditions for a quadratic equation to have real and distinct roots are inherently connected to the discriminant. As we identified, a positive discriminant indicates two real and distinct roots. Moreover, from calculus and algebra, we understand that the roots of a function are the values where the function intersects the x-axis. Distinct real roots mean the graph of the quadratic function touches the x-axis at two points.

For the equation provided, not only do we require real and distinct roots, but an additional boundary condition is set – both roots must be greater than or equal to 4. This translates into determining a value of k where the vertex of the parabola, defined by the equation, is at or above the horizontal line y = 4. The combination of a positive discriminant and the application of Vieta’s formulas guides us to conclude that k must be greater than 1 to satisfy all given conditions.