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Simplify (5p^(2))/(p^(2)+2p-15)*(p^(2)-25)/(25 p)

Simplify 5p2p2+2p−15⋅p2−2525p \frac{5 p^{2}}{p^{2}+2 p-15} \cdot \frac{p^{2}-25}{25 p} \newline

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Full solution

Q. Simplify 5p2p2+2p−15⋅p2−2525p \frac{5 p^{2}}{p^{2}+2 p-15} \cdot \frac{p^{2}-25}{25 p} \newline
  1. Factor Quadratic Expressions: Factor the quadratic expressions where possible.\newlineThe quadratic expression p2+2p−15p^2 + 2p - 15 can be factored into (p+5)(p−3)(p + 5)(p - 3) because (p+5)(p−3)=p2−3p+5p−15=p2+2p−15(p + 5)(p - 3) = p^2 - 3p + 5p - 15 = p^2 + 2p - 15.\newlineThe quadratic expression p2−25p^2 - 25 can be factored into (p+5)(p−5)(p + 5)(p - 5) because (p+5)(p−5)=p2−5p+5p−25=p2−25(p + 5)(p - 5) = p^2 - 5p + 5p - 25 = p^2 - 25.
  2. Rewrite with Factored Forms: Rewrite the original expression with the factored forms.\newlineThe expression becomes (5p2)/((p+5)(p−3))×((p+5)(p−5))/(25p)(5p^2)/((p + 5)(p - 3)) \times ((p + 5)(p - 5))/(25p).
  3. Cancel Common Factors: Cancel out common factors.\newlineWe can cancel out the common factor of (p+5)(p + 5) from the numerator of the first fraction and the numerator of the second fraction.\newlineWe can also cancel out pp from 5p25p^2 in the numerator of the first fraction and 25p25p in the denominator of the second fraction.\newlineThe expression now simplifies to 5pp−3×p−525\frac{5p}{p - 3} \times \frac{p - 5}{25}.
  4. Simplify Further: Simplify the expression further.\newlineWe can cancel out the common factor of 55 from 5p5p in the numerator and 2525 in the denominator.\newlineThe expression now simplifies to pp−3×p−55\frac{p}{p - 3} \times \frac{p - 5}{5}.
  5. Multiply Remaining Expressions: Multiply the remaining expressions. Multiplying the numerators together and the denominators together, we get (p×(p−5))/((p−3)×5)(p \times (p - 5))/((p - 3) \times 5).
  6. Expand Numerator: Expand the numerator.\newlineExpanding the numerator, we get p2−5pp^2 - 5p.\newlineThe expression now is (p2−5p)/((p−3)⋅5)(p^2 - 5p)/((p - 3) \cdot 5).
  7. Leave Denominator Factored: Leave the denominator in factored form.\newlineThe final simplified expression is (p2−5p)/(5(p−3))(p^2 - 5p)/(5(p - 3)).

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