Answer:
The zeros of the function:
[tex]x=5.53 ,[/tex] [tex]x=3.22[/tex]
Step-by-step explanation:
Given the function
[tex]f\left(x\right)=2x^2-17.5x+35.6[/tex]
To find zeros, put f(x)=0
[tex]0=2x^2-17.5x+35.6[/tex]
[tex]\mathrm{Multiply\:both\:sides\:by\:}10[/tex]
[tex]0\cdot \:10=2x^2\cdot \:10-17.5x\cdot \:10+35.6\cdot \:10[/tex]
[tex]0=20x^2-175x+356[/tex]
switch sides
[tex]20x^2-175x+356=0[/tex]
subtract 356 from both sides
[tex]20x^2-175x+356-356=0-356[/tex]
[tex]20x^2-175x=-356[/tex]
divide both sides by 20
[tex]\frac{20x^2-175x}{20}=\frac{-356}{20}[/tex]
[tex]x^2-\frac{35x}{4}=-\frac{89}{5}[/tex]
[tex]\mathrm{Add\:}a^2=\left(-\frac{35}{8}\right)^2\mathrm{\:to\:both\:sides}[/tex]
[tex]x^2-\frac{35x}{4}+\left(-\frac{35}{8}\right)^2=-\frac{89}{5}+\left(-\frac{35}{8}\right)^2[/tex]
applying perfect square
[tex]\left(x-\frac{35}{8}\right)^2=\frac{429}{320}[/tex]
[tex]\mathrm{For\:}f^2\left(x\right)=a\mathrm{\:the\:solutions\:are\:}f\left(x\right)=\sqrt{a},\:-\sqrt{a}[/tex]
solving
[tex]x-\frac{35}{8}=\sqrt{\frac{429}{320}}[/tex]
[tex]x=\frac{\sqrt{2145}}{40}+\frac{35}{8}[/tex]
[tex]x=5.53[/tex]
also solving
[tex]x-\frac{35}{8}=-\sqrt{\frac{429}{320}}[/tex]
[tex]x=-\frac{\sqrt{2145}}{40}+\frac{35}{8}[/tex]
[tex]x=3.22[/tex]
[tex]\mathrm{The\:solutions\:to\:the\:quadratic\:equation\:are:}[/tex]
[tex]x=5.53 ,[/tex] [tex]x=3.22[/tex]
Therefore, the zeros of the function:
[tex]x=5.53 ,[/tex] [tex]x=3.22[/tex]
