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Solve the rational equation 4 divided by x equals quantity 3 times x plus 2 end quantity divided by x squared, and check for extraneous solutions.

Solve the rational equation 4 divided by x equals quantity 3 times x plus 2 end quantity divided by x squared, and check for extraneous solutions. class=

Answer :

Space

Answer:

x = 2

Step-by-step explanation:

Step 1: Write equation

[tex]\frac{4}{x}=\frac{3x+2}{x^2}[/tex]

Step 2: Solve for x

  1. Cross multiply: [tex]4x^2=x(3x+2)[/tex]
  2. Distribute x: [tex]4x^2=3x^2+2x[/tex]
  3. Subtract 3x² on both sides: [tex]x^2=2x[/tex]
  4. Subtract 2x on both sides: [tex]x^2-2x=0[/tex]
  5. Factor: [tex]x(x-2)=0[/tex]
  6. Solve: [tex]x = 0, 2[/tex]

Step 3: Check

Plug in x to verify it's a solution.

  1. Substitute: [tex]\frac{4}{0}=\frac{3(0)+2}{0^2}[/tex]
  2. Evaluate: undef (cannot divide by 0)

  1. Substitute: [tex]\frac{4}{2}=\frac{3(2)+2}{2^2}[/tex]
  2. Evaluate: [tex]2=\frac{6+2}{4}[/tex]
  3. Evaluate: [tex]2=\frac{8}{4}[/tex]
  4. Evaluate: [tex]2=2[/tex]

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