Solving for k: Quadratic Equation with Discriminant

This problem asks us to find the value of the constant k in a given quadratic equation, using the information that its discriminant is 9.

Understanding the Discriminant of a Quadratic Equation

A quadratic equation is generally written as $ax^2 + bx + c = 0$. The discriminant, denoted by $\Delta$, is calculated using the formula:

$$ \Delta = b^2 - 4ac $$

The value of the discriminant helps determine the nature of the roots (solutions) of the quadratic equation.

Applying the Formula to the Given Equation

The equation provided is $x^2 + 3x + k = 0$. Let's identify the coefficients:

• The coefficient 'a' (for $x^2$) is 1.
• The coefficient 'b' (for $x$) is 3.
• The coefficient 'c' (the constant term) is k.

We are given that the discriminant ($\Delta$) is 9.

Step-by-Step Calculation to Find k

Using the discriminant formula $\Delta = b^2 - 4ac$, we substitute the values we know:

$$ 9 = (3)^2 - 4(1)(k) $$

First, calculate $3^2$:

$$ 9 = 9 - 4(1)(k) $$

Simplify the term $4(1)(k)$:

$$ 9 = 9 - 4k $$

Now, we need to isolate k. Subtract 9 from both sides of the equation:

$$ 9 - 9 = 9 - 4k - 9 $$

This simplifies to:

$$ 0 = -4k $$

Finally, divide both sides by -4 to solve for k:

$$ \frac{0}{-4} = \frac{-4k}{-4} $$

$$ 0 = k $$

Therefore, the value of k is 0.

Final Answer Verification

If we substitute $k=0$ back into the original equation, we get $x^2 + 3x = 0$. The discriminant is $3^2 - 4(1)(0) = 9 - 0 = 9$, which matches the given information.