Answer:
[tex]x=\dfrac{-17+ \sqrt{451} }{18}, \quad x=\dfrac{-17- \sqrt{451} }{18}[/tex]
Step-by-step explanation:
[tex]\textsf{Given equation}:[/tex]
[tex]\dfrac{2}{3}x+\dfrac{3x}{x}=\dfrac{4x}{2}(3x+6)[/tex]
[tex]\textsf{Cancel the common factor } x \textsf{ in }\dfrac{3x}{x}:[/tex]
[tex]\implies \dfrac{2}{3}x+3=\dfrac{4x}{2}(3x+6)[/tex]
[tex]\textsf{Simplify }\dfrac{4x}{2}\textsf{ by dividing the numbers}:[/tex]
[tex]\implies \dfrac{2}{3}x+3=2x(3x+6)[/tex]
[tex]\textsf{Apply the distributive law}\quad \:a\left(b+c\right)=ab+ac:[/tex]
[tex]\implies \dfrac{2}{3}x+3=6x^2+12x[/tex]
[tex]\textsf{Multiply both sides by 3 to cancel the denominator of }\dfrac{2}{3}:[/tex]
[tex]\implies \dfrac{2}{3}x\cdot 3+3\cdot 3=6x^2\cdot 3+12x \cdot 3[/tex]
[tex]\implies 2x+9=18x^2+36x[/tex]
[tex]\textsf{Switch sides}:[/tex]
[tex]\implies 18x^2+36x=2x+9[/tex]
[tex]\textsf{Subtract }2x \textsf{ from both sides}:[/tex]
[tex]\implies 18x^2+36x-2x=2x+9-2x[/tex]
[tex]\implies 18x^2+34x=9[/tex]
[tex]\textsf{Subtract 9 from both sides}:[/tex]
[tex]\implies 18x^2+34x-9=9-9[/tex]
[tex]\implies 18x^2+34x-9=0[/tex]
Solve using the Quadratic Formula:
[tex]x=\dfrac{-b \pm \sqrt{b^2-4ac} }{2a}\quad\textsf{when }\:ax^2+bx+c=0[/tex]
Define the variables:
[tex]\implies a=18, \quad b=34, \quad c=-9[/tex]
Substitute the defined variables into the quadratic formula and solve for x:
[tex]\implies x=\dfrac{-34 \pm \sqrt{34^2-4(18)(-9)} }{2(18)}[/tex]
[tex]\implies x=\dfrac{-34 \pm \sqrt{1156+648} }{36}[/tex]
[tex]\implies x=\dfrac{-34 \pm \sqrt{1804} }{36}[/tex]
[tex]\implies x=\dfrac{-34 \pm \sqrt{4 \cdot 451} }{36}[/tex]
[tex]\implies x=\dfrac{-34 \pm \sqrt{4}\sqrt{451} }{36}[/tex]
[tex]\implies x=\dfrac{-34 \pm 2\sqrt{451} }{36}[/tex]
[tex]\implies x=\dfrac{-17 \pm \sqrt{451} }{18}[/tex]
Therefore, the solutions to the given equation are:
[tex]x=\dfrac{-17+ \sqrt{451} }{18}, \quad x=\dfrac{-17- \sqrt{451} }{18}[/tex]
Learn more about the Quadratic Formula here:
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