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The question is:

Rectangular floor mats have an area of $x^2 + 2x - 15 \text{ cm}^2$. The length is $x + 5\text{ cm}$, and the width is $100\text{ cm}$. How do I find out the value of $x$, and from there the length of the mat? The equation to find the width is $$\frac{x^2+2x-15}{\text{length}}$$

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  • $\begingroup$ Can you factor $x^2+2x-15$? $\endgroup$ – Ghartal Jul 4 '16 at 8:08
  • $\begingroup$ Area of rectangle $=$ length $\times$ width. Use that and solve the quadratic equation. $\endgroup$ – John_dydx Jul 4 '16 at 8:09
  • $\begingroup$ There isn't a quadratic equation involved in this I believe John $\endgroup$ – Kiwi Jul 4 '16 at 8:12
  • $\begingroup$ @Kiwi You are right, see my hint. $\endgroup$ – Olivier Oloa Jul 4 '16 at 8:16
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Hint. One may solve, for $x>0$, $$ \frac{x^2+2x-15}{x+5}=100 $$ or $$ \frac{(x+5)(x-3)}{x+5}=100 $$

Can you take it from here?

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  • $\begingroup$ Ah, so the (x+5)'s cancel out, meaning x-3=100, so x is 103, meaning length is 108. Thanks so much!! $\endgroup$ – Kiwi Jul 4 '16 at 8:14
  • $\begingroup$ @Kiwi We have $x^2+2x-15=(x^2+2x+1)-16=(x+1)^2-4^2=[(x+1)-4][(x+1)+4]=(x-3)(x+5)$. $\endgroup$ – Olivier Oloa Jul 4 '16 at 8:20

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