Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 7 - Section 7.2 - Systems of Linear Equations in Three Variables - Exercise Set - Page 829: 21


The quadratic equation is $2{{x}^{2}}+x-5$.

Work Step by Step

The standard quadratic equation is shown below: $y=a{{x}^{2}}+bx+c$ The points are $\left( -1,-4 \right),\left( 1,-2 \right),\left( 2,5 \right)$. In order to get the quadratic function, find the values of a, b, and c, and substitute the values of the ordered pairs in the quadratic function. Put $x=-1$ and $y=-4$ $\begin{align} & -4=a{{\left( -1 \right)}^{2}}+b\left( -1 \right)+c \\ & -4=a-b+c \end{align}$ (I) Put $x=1$ and $y=-2$ $\begin{align} & -2=a{{\left( 1 \right)}^{2}}+b\left( 1 \right)+c \\ & -2=a+b+c \end{align}$ (II) Put $x=2$ and $y=5$ $\begin{align} & 5=a{{\left( 2 \right)}^{2}}+b\left( 2 \right)+c \\ & 5=4a+2b+c \end{align}$ (III) Get the system of equations as follows: $\begin{align} & a-b+c=-4 \\ & a+b+c=-2 \\ & 4a+2b+c=5 \end{align}$ Eliminate a from equations (I) and (II) to get; $\begin{align} & \text{ }a-b+c=-4 \\ & -a-b-c=2 \\ \end{align}$ $-2b=-2$ Divide both sides by $-2$ to get: $\begin{align} & \frac{-2b}{-2}=\frac{-2}{-2} \\ & b=1 \end{align}$ Putting in the value of b in equation (I): we get $\begin{align} & a-\left( 1 \right)+c=-4 \\ & a+c-1=-4 \end{align}$ Now, adding 1 to both sides to get: $\begin{align} & a+c-1+1=-4+1 \\ & a+c=-3 \end{align}$ (IV) Putting the value of b in equation (III) to get: $\begin{align} & 4a+2\left( 1 \right)+c=5 \\ & 4a+2+c=5 \end{align}$ Subtract 2 from both sides to get, $\begin{align} & 4a+c+2-2=5-2 \\ & 4a+c=3 \end{align}$ (V) By eliminating c from equations (IV) and (V) $\begin{align} & 4a+c=\text{3} \\ & -a-c=\text{3} \end{align}$ $3a\text{ }=\text{ }6$ (VI) By dividing both sides by 3: we get, $\begin{align} & \frac{3a}{3}=\frac{6}{3} \\ & a=2 \end{align}$ Putting the value of a in equation (IV) to find the value of c: we get $2+c=-3$ Now, subtract 2 from both sides to get, $\begin{align} & 2+c-2=-3-2 \\ & c=-5 \end{align}$ Now putting the values of a, b and c to get the equation $2{{x}^{2}}+x-5$. Hence, the quadratic equation is $2{{x}^{2}}+x-5$.
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