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Answer:

m = 6 or m = 3/2

Step-by-step explanation:

Solve for m over the real numbers:

3 m = 2 (m - 3)^2

Expand out terms of the right hand side:

3 m = 2 m^2 - 12 m + 18

Subtract 2 m^2 - 12 m + 18 from both sides:

-2 m^2 + 15 m - 18 = 0

The left hand side factors into a product with three terms:

-(m - 6) (2 m - 3) = 0

Multiply both sides by -1:

(m - 6) (2 m - 3) = 0

Split into two equations:

m - 6 = 0 or 2 m - 3 = 0

Add 6 to both sides:

m = 6 or 2 m - 3 = 0

Add 3 to both sides:

m = 6 or 2 m = 3

Divide both sides by 2:

Answer: m = 6 or m = 3/2

First of all, let's expand the square on the right hand side:

[tex]3m=2(m^2-6m+9)[/tex]

Distribute the two on the right hand side:

[tex]3m=2m^2-12m+18[/tex]

Subtract 3m from both sides:

[tex]0=2m^2-15m+18[/tex]

Use the quadratic formula to solve:

[tex]m_{1,2}=\dfrac{15\pm\sqrt{225-144}}{4}=\dfrac{15\pm 9}{4}[/tex]

So, the solutions are

[tex]m_1=\dfrac{15+9}{4}=\dfrac{24}{4}=6[/tex]

[tex]m_2=\dfrac{15-9}{4}=\dfrac{6}{4}=\dfrac{3}{2}[/tex]

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