# What is the vertex form of 7y=3x^2+2x − 4.?

Feb 4, 2016

$y = \frac{3}{7} {\left(x + \frac{1}{3}\right)}^{2} - \frac{13}{21}$

#### Explanation:

write as:

$y = \frac{3}{7} {x}^{2} + \frac{2}{7} x - \frac{4}{7}$..................................(1)

$y = \frac{3}{7} \left({x}^{2} + \textcolor{b l u e}{\frac{2}{3} x}\right) - \frac{4}{7}$

consider the $\frac{2}{3} \text{ from "color(blue)( 2/3x)" and multiply it by } \textcolor{b r o w n}{\frac{1}{2}}$

$\textcolor{b r o w n}{\frac{1}{2}} \times \textcolor{b l u e}{\frac{2}{3}} = \textcolor{g r e e n}{\frac{1}{3}}$

$y \ne \frac{3}{7} {\left(x + \textcolor{g r e e n}{\frac{1}{3}}\right)}^{2} - \frac{4}{7} \text{ }$$\textcolor{p u r p \le}{\text{ This introduces an error!}}$

Let $k$ be some constant then:

$y = \frac{3}{7} {\left(x + \frac{1}{3}\right)}^{2} + k - \frac{4}{7}$ ...................(2) $\textcolor{p u r p \le}{\text{Corrected the error!}}$

expanding to find the value of k
$y = \frac{3}{7} {x}^{2} + \frac{2}{7} x + \frac{1}{21} + k - \frac{4}{7}$ ......................(3)
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Equate equation (1) to equation (3) through y

$\cancel{\frac{3}{7} {x}^{2}} + \cancel{\frac{2}{7} x} - \cancel{\frac{4}{7}} \text{ "=" } \cancel{\frac{3}{7} {x}^{2}} + \cancel{\frac{2}{7} x} + \frac{1}{21} + k - \cancel{\frac{4}{7}}$

$k + \frac{1}{21} = 0 \text{ "->" } k = - \frac{1}{21}$.............................(4)
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Substitute (4) into (2) giving

$y = \frac{3}{7} {\left(x + \frac{1}{3}\right)}^{2} - \frac{1}{21} - \frac{4}{7} \ldots \ldots \ldots \ldots \ldots \ldots . \left({2}_{a}\right)$

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Solution:

$y = \frac{3}{7} {\left(x + \frac{1}{3}\right)}^{2} - \frac{13}{21}$

$\textcolor{p u r p \le}{\text{Please check the arithmetic. I can not spot any error but I am not}}$ $\textcolor{p u r p \le}{\text{totally satisfied with the answer!}}$