Question

find all values of k for which the quadratic equation has one real solution.
$7x^2 + 9x + k = 0$
write your answer as an equality or inequality in terms of k.

Explanation:

Step1: Recall Discriminant Formula

For a quadratic equation \(ax^2 + bx + c = 0\), the discriminant is \(D = b^2 - 4ac\). If the equation has one real solution, \(D = 0\).

Step2: Identify Coefficients

In \(7x^2 + 9x + k = 0\), \(a = 7\), \(b = 9\), \(c = k\).

Step3: Set Discriminant to Zero

Substitute into discriminant formula: \(9^2 - 4\times7\times k = 0\).

Step4: Solve for k

Calculate \(81 - 28k = 0\). Then, \(28k = 81\), so \(k=\frac{81}{28}\).

Answer:

\(k = \frac{81}{28}\)