Solving for k in the Quadratic Equation 9x^2 8kx 16 0 to Ensure Equal Roots

For a quadratic equation to have equal roots, the discriminant must be equal to zero. This article explains how to solve for the value of k in the equation 9x^2 8kx 16 0 to ensure that the roots are equal. We will delve into the concept, the formula, and the detailed steps required to find the correct value of k.

Understanding the Concept of Equal Roots

In a quadratic equation of the form ax^2 bx c 0, the roots are equal when the discriminant (D) is zero. The discriminant is given by the formula:

D b^2 - 4ac

Application to the Given Equation

Given the quadratic equation 9x^2 8kx 16 0, we identify the coefficients as:

a 9 b 8k c 16

To ensure that the roots are equal, we set the discriminant equal to zero:

D b^2 - 4ac 0

Calculating the Discriminant

Substituting the values of b, a, and c into the discriminant formula:

D (8k)^2 - 4(9)(16)

This simplifies to:

64k^2 - 576 0

Solving the Equation for k

To solve for k, we first isolate k^2 on one side of the equation:

64k^2 576

Divide both sides by 64:

k^2 9

Taking the square root of both sides, we have:

k ±3

Final Answer

The values of k for which the equation 9x^2 8kx 16 0 has equal roots are:

boxed{k 3} and boxed{k -3}

Summary of the Key Points

To find the value of k in the quadratic equation, we need to ensure the discriminant is zero. The discriminant in the equation 9x^2 8kx 16 0 is 64k^2 - 576. Setting the discriminant to zero, we solve for k and find that k ±3.

Additional Information

Understanding the discriminant is crucial in solving quadratic equations. The discriminant tells us the nature of the roots:

If D 0, there are two distinct real roots. If D 0, there are exactly two real roots (or equal roots). If D 0, there are no real roots (the roots are complex).

By mastering the discriminant and how to apply it, you can solve a wide range of quadratic equations efficiently and accurately.