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: 25

Answer

$y=2x^2+x-5$.

Work Step by Step

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