To determine the zeros of the function [tex]\( f(x) = 3x^2 - 7x + 1 \)[/tex], you can use the quadratic formula, which is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are the coefficients of the quadratic equation [tex]\( ax^2 + bx + c \)[/tex].
For the quadratic function [tex]\( f(x) = 3x^2 - 7x + 1 \)[/tex]:
- [tex]\( a = 3 \)[/tex]
- [tex]\( b = -7 \)[/tex]
- [tex]\( c = 1 \)[/tex]
Step 1: Calculate the discriminant ([tex]\( \Delta \)[/tex]):
[tex]\[ \Delta = b^2 - 4ac \][/tex]
[tex]\[ \Delta = (-7)^2 - 4 \cdot 3 \cdot 1 \][/tex]
[tex]\[ \Delta = 49 - 12 \][/tex]
[tex]\[ \Delta = 37 \][/tex]
The discriminant ([tex]\( \Delta \)[/tex]) is 37.
Step 2: Calculate the two roots using the quadratic formula:
[tex]\[ x_1 = \frac{-b + \sqrt{\Delta}}{2a} \][/tex]
[tex]\[ x_1 = \frac{-(-7) + \sqrt{37}}{2 \cdot 3} \][/tex]
[tex]\[ x_1 = \frac{7 + \sqrt{37}}{6} \][/tex]
and
[tex]\[ x_2 = \frac{-b - \sqrt{\Delta}}{2a} \][/tex]
[tex]\[ x_2 = \frac{-(-7) - \sqrt{37}}{2 \cdot 3} \][/tex]
[tex]\[ x_2 = \frac{7 - \sqrt{37}}{6} \][/tex]
So, the two roots are:
[tex]\[ x_1 \approx 2.18046042171637 \][/tex]
[tex]\[ x_2 \approx 0.15287291161696345 \][/tex]
Therefore, the zeros of the function [tex]\( f(x) = 3x^2 - 7x + 1 \)[/tex] are approximately [tex]\( x_1 = 2.18046042171637 \)[/tex] and [tex]\( x_2 = 0.15287291161696345 \)[/tex].
