To determine the zeros of the quadratic function [tex]\( f(x) = 5.23x^2 - 4.68x - 1.93 \)[/tex], we use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, the coefficients are:
- [tex]\( a = 5.23 \)[/tex]
- [tex]\( b = -4.68 \)[/tex]
- [tex]\( c = -1.93 \)[/tex]
Plug these values into the quadratic formula:
1. Calculate the discriminant:
[tex]\[ \text{Discriminant} = b^2 - 4ac \][/tex]
[tex]\[ \text{Discriminant} = (-4.68)^2 - 4 \cdot 5.23 \cdot (-1.93) \][/tex]
2. Simplify the discriminant:
[tex]\[ \text{Discriminant} = 21.9024 + 40.3164 \][/tex]
[tex]\[ \text{Discriminant} = 62.2188 \][/tex]
3. Take the square root of the discriminant:
[tex]\[ \sqrt{62.2188} \][/tex]
4. Apply the quadratic formula for the solutions [tex]\( x \)[/tex]:
[tex]\[ x = \frac{-(-4.68) \pm \sqrt{62.2188}}{2 \cdot 5.23} \][/tex]
[tex]\[ x = \frac{4.68 \pm \sqrt{62.2188}}{10.46} \][/tex]
5. Calculate the two solutions:
[tex]\[ x_1 = \frac{4.68 + \sqrt{62.2188}}{10.46} \][/tex]
[tex]\[ x_2 = \frac{4.68 - \sqrt{62.2188}}{10.46} \][/tex]
Rounding these solutions to three decimal places, we get:
[tex]\[ x_1 \approx 1.202 \][/tex]
[tex]\[ x_2 \approx -0.307 \][/tex]
Thus, the zeros of the function are:
[tex]\[ 1.202, -0.307 \][/tex]
