Solve y = x² + 8x + 20: Finding Values Where Function is Negative

Look at the following function:

$y=x^2+8x+20$

Determine for which values of $x$ the following is true:

$f(x) < 0$

To identify for which values of $x$ the function $y = x^2 + 8x + 20$ is negative, we will analyze the quadratic equation:

• Step 1: Identify the coefficients: $a = 1$, $b = 8$, $c = 20$.
• Step 2: Calculate the discriminant $\Delta = b^2 - 4ac$.
• Step 3: Evaluate whether the parabola intersects the x-axis based on the discriminant.

Calculating the discriminant:
$\Delta = 8^2 - 4 \cdot 1 \cdot 20 = 64 - 80 = -16$.

The discriminant $\Delta = -16$ is less than zero, which means there are no real roots. The parabola does not intersect the x-axis and opens upwards because the coefficient a is positive.

Therefore, the values of $y = x^2 + 8x + 20$ are always greater than zero for all real $x$. The quadratic function does not take negative values for any real $x$.

The correct answer is: The function has no negative values.