Quadratic Equation Analysis

The given equation is a quadratic equation of the form $ax^2 + bx + c = 0$. In this case, the equation is $x^2 + kx + 361 = 0$. Comparing the terms, we have:

• $a = 1$
• $b = k$
• $c = 361$

Condition for Equal Roots

A quadratic equation has equal roots if its discriminant ($\Delta$) is equal to zero. The discriminant is calculated using the formula:

$ \Delta = b^2 - 4ac $

For equal roots, we set $\Delta = 0$:

$ b^2 - 4ac = 0 $

Solving for k

Substitute the values of $a$, $b$, and $c$ into the condition:

$ k^2 - 4(1)(361) = 0 $

Simplify the equation:

$ k^2 - 1444 = 0 $

Isolate $k^2$:

$ k^2 = 1444 $

Take the square root of both sides to find the values of $k$:

$ k = \pm \sqrt{1444} $

Calculate the square root:

$ \sqrt{1444} = 38 $

Therefore, the values of $k$ for which the equation will have equal roots are:

$ k = \pm 38 $