Question

solve for ( v ).
( 4 - \frac{3}{v + 2} = -\frac{3}{v + 3} )
if there is more than one solution, separate them with commas.
if there is no solution, click on
o solution\.
( v = square )

Explanation:

Step1: Eliminate denominators

Multiply all terms by $(v+2)(v+3)$:
$$4(v+2)(v+3) - 3(v+3) = -3(v+2)$$

Step2: Expand all expressions

Expand each product using distribution:
$$4(v^2+5v+6) - 3v -9 = -3v -6$$
$$4v^2+20v+24 -3v -9 = -3v -6$$

Step3: Simplify both sides

Combine like terms on left side:
$$4v^2+17v+15 = -3v -6$$

Step4: Move all terms to left

Set equation to 0 by adding $3v+6$:
$$4v^2+20v+21 = 0$$

Step5: Solve quadratic equation

Use quadratic formula $v=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ where $a=4, b=20, c=21$:
$$v=\frac{-20\pm\sqrt{20^2-4(4)(21)}}{2(4)}$$
$$v=\frac{-20\pm\sqrt{400-336}}{8}$$
$$v=\frac{-20\pm\sqrt{64}}{8}$$
$$v=\frac{-20\pm8}{8}$$

Step6: Calculate two solutions

Compute each root separately:
$$v_1=\frac{-20+8}{8}=\frac{-12}{8}=-\frac{3}{2}$$
$$v_2=\frac{-20-8}{8}=\frac{-28}{8}=-\frac{7}{2}$$

Step7: Verify no extraneous roots

Check that $v
eq-2,-3$ (both solutions satisfy this).

Answer:

$v = -\frac{7}{2}, -\frac{3}{2}$