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We're learning about Quadratics, but I'm not exactly sure how this applies to it:

$\dfrac{w + h}{w} = \dfrac{w}{h}$. If the height is 16 inches, what is its width? (Round to the nearest tenth.)

Can someone help me out?

N. F. Taussig
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johny
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2 Answers2

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Plug in $h=16$ and multiply both sides by $16w$ to have $$ 16(w+16)=w^2 $$

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If we substitute $16$ for $h$ in the expression $$\frac{w + h}{w} = \frac{w}{h}$$ we obtain $$\frac{w + 16}{w} = \frac{w}{16}$$ If we multiply both sides of the equation by $16w$ (or cross-multiply), we obtain \begin{align*} 16(w + 16) & = w^2\\ 16w + 256 & = w^2\\ 0 & = w^2 - 16w - 256 \end{align*} which is a quadratic equation in $w$, which you can solve by completing the square of applying the Quadratic Formula.

N. F. Taussig
  • 76,571
  • The "solve" part is what I don't get. w + 16 / w = w / 16? – johny Mar 09 '15 at 21:53
  • Alright, I understand after you get the 16(w + 16) = w^2 part. Where do you cross multiply though? – johny Mar 09 '15 at 22:01
  • Inclusively, don't you have to write it as 0 = -w^2 instead of w^2 since you subtracted it? – johny Mar 09 '15 at 22:02
  • Observe that I subtracted $16w + 256$ from each side of the equation rather than $w^2$. When I transformed the equation from $$\frac{w + 16}{w} = \frac{w}{16}$$ I cross-multiplied since I multiplied the numerator of the fraction on the LHS by the denominator of the fraction on the RHS and set it equal to the product of the denominator on the LHS and the numerator on the RHS using the rule $$\frac{a}{b} = \frac{c}{d} \Longleftrightarrow ad = bc$$ (assuming $b, d \neq 0$). – N. F. Taussig Mar 09 '15 at 22:06
  • How come you didn't subtract the w^2? Should that always be positive? – johny Mar 09 '15 at 22:09
  • If you subtract $w^2$, you would obtain $$-w^2 + 16w + 256 = 0$$ which is equivalent to the expression I obtained. I chose to subtract $16w + 256$ instead so that the quadratic would have the form $x^2 + px + q$, which is easier to solve by completing the square or by using the Quadratic Formula. – N. F. Taussig Mar 09 '15 at 22:12