To find the zeros of the quadratic function f(x) = 9x² + 54x + 19, we can use the quadratic formula:
x = (-b ± √(b² – 4ac)) / (2a)
In this equation, a, b, and c are the coefficients from the standard form of a quadratic equation, which is ax² + bx + c = 0.
For our function, we have:
- a = 9
- b = 54
- c = 19
Now, we can substitute these values into the quadratic formula:
x = ( -54 ± √(54² – 4 * 9 * 19)) / (2 * 9)
Calculating the discriminant:
54² = 2916
4 * 9 * 19 = 684
So, the discriminant b² – 4ac = 2916 – 684 = 2232.
Now substituting back into the formula:
x = (-54 ± √2232) / 18
Next, we need to calculate √2232:
√2232 ≈ 47.23 (rounded to two decimal places)
Now we can find the two possible values for x:
x₁ = (-54 + 47.23) / 18 ≈ -0.37
x₂ = (-54 – 47.23) / 18 ≈ -5.67
Therefore, the zeros of the quadratic function f(x) = 9x² + 54x + 19 are approximately:
- x ≈ -0.37
- x ≈ -5.67
These values represent the points where the graph of the quadratic function intersects the x-axis.