Intermediate Algebra for College Students (7th Edition)

Published by Pearson
ISBN 10: 0-13417-894-7
ISBN 13: 978-0-13417-894-3

Chapter 3 - Section 3.3 - Systems of Linear Equations in Three Variables - Exercise Set - Page 216: 26

Answer

$y=x^2-6x+8$.

Work Step by Step

The given points are $(1,3),(3,-1),(4,0)$ The quadratic function is $y=ax^2+bx+c$. Plug $(x,y)=(1,3)$. $\Rightarrow 3=a(1)^2+b(1)+c$ Simplify. $\Rightarrow 3=a+b+c$ ...... (1) Plug $(x,y)=(3,-1)$. $\Rightarrow -1=a(3)^2+b(3)+c$ Simplify. $\Rightarrow -1=9a+3b+c$ ...... (2) Plug $(x,y)=(4,0)$. $\Rightarrow 0=a(4)^2+b(4)+c$ Simplify. $\Rightarrow 0=16a+4b+c$ ...... (3) Subtract equation (1) from equation (3). $\Rightarrow 0-3=16a+4b+c-(a+b+c)$ Simplify. $\Rightarrow -3=16a+4b+c-a-b-c$ $\Rightarrow -3=15a+3b$ Divide both sides by $3$. $\Rightarrow -1=5a+b$ ...... (4) Subtract equation (2) from equation (3). $\Rightarrow 0-(-1)=16a+4b+c-(9a+3b+c)$ Simplify. $\Rightarrow 1=16a+4b+c-9a-3b-c$ $\Rightarrow 1=7a+b$ ...... (5) Subtract equation (4) from equation (5). $\Rightarrow -1-1=5a+b-(7a+b)$ Simplify. $\Rightarrow -2=5a+b-7a-b$ $\Rightarrow -2=-2a$ Divide both sides by $-2$. $\Rightarrow 1=a$ Substitute the value of $a$ into equation (4). $\Rightarrow -1=5(1)+b$ $\Rightarrow -1=5+b$ Subtract $5$ from both sides. $\Rightarrow -1-5=5+b-5$ Simplify. $\Rightarrow -6=b$ Substitute the values of $a$ and $b$ into equation (2) $\Rightarrow -1=9(1)+3(-6)+c$ ...... (2) Simplify. $\Rightarrow -1=9-18+c$ $\Rightarrow -1=-9+c$ Add $9$ to both sides. $\Rightarrow -1+9=-9+c+9$ Simplify. $\Rightarrow 8=c$ Plug all values into the quadratic equation. $\Rightarrow y=1x^2+(-6)x+8$. $\Rightarrow y=x^2-6x+8$.
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