How do you solve $3 x^{2} + 5 x = 2$ $3x^2+5x=2$ by completing the square?

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How do you solve $3 x^{2} + 5 x = 2$ $3x^2+5x=2$ by completing the square?

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Answer 1 · 2016-10-06T03:00:14

$3 {(x + \frac{5}{6})}^{2} - \frac{49}{12} = y$ $3(x+5/6)^2-49/12=y$

Explanation:

The goal of completing the square is to convert the equation into a perfect square trinomial. The benefit of this form is to be able to identify the vertex of a parabola very easily.

Factor out a 3

$3 (x^{2} + \frac{5}{3} x) = 2$ $3(x^2+5/3x)=2$

Take the coefficient from the x term and divide it by 2 and then square it.

${(\frac{\frac{5}{3}}{2})}^{2} = {(\frac{5}{3} \cdot \frac{1}{2})}^{2} = {(\frac{5}{6})}^{2} = \frac{25}{36}$ $((5/3)/2)^2=(5/3*1/2)^2=(5/6)^2=25/36$

Include $\frac{25}{36}$ $25/36$ on the left and include $3 (\frac{25}{36})$ $3(25/36)$ on the right because we factored out a 3 on the left in an earlier step.

$3 (x^{2} + \frac{5}{3} x + \frac{25}{36}) = 2 + 3 (\frac{25}{36})$ $3(x^2+5/3x+25/36)=2+3(25/36)$

We now have a perfect square trinomial that can be written in a more compact form

$3 {(x + \frac{5}{6})}^{2} = 2 + 3 (\frac{25}{36})$ $3(x+5/6)^2=2+3(25/36)$

Simplify

$3 {(x + \frac{5}{6})}^{2} = 2 + 3 (\frac{25}{3612})$ $3(x+5/6)^2=2+cancel3(25/(cancel36 12))$

$3 {(x + \frac{5}{6})}^{2} = 2 + (\frac{25}{12})$ $3(x+5/6)^2=2+(25/(12))$

Convert $2$ $2$ to $\frac{24}{12}$ $24/12$ so that we have common denominator.

$3 {(x + \frac{5}{6})}^{2} = \frac{24}{12} + (\frac{25}{12})$ $3(x+5/6)^2=24/12+(25/(12))$

$3 {(x + \frac{5}{6})}^{2} = \frac{49}{12}$ $3(x+5/6)^2=49/12$

Subtract $\frac{49}{12}$ $49/12$

$3 {(x + \frac{5}{6})}^{2} - \frac{49}{12} = 0$ $3(x+5/6)^2-49/12=0$

$3 {(x + \frac{5}{6})}^{2} - \frac{49}{12} = y$ $3(x+5/6)^2-49/12=y$

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