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Using Polynomials to Solve Word Problems

Two sisters are mowing a rectangular lawn with the dimensions $\mathrm{100ft \times 40ft}$ . One sister starts by mowing a wide strip around the outside and the other finishes by mowing the small rectangle that is left. How wide should the first sister’s strip be if they are to share the work equally?

$\mathrm{Area_{B} = (100-2x)(40-2x)}$
$\mathrm{Area_{A}=(100)(40)-Area_{B}}$

$\mathrm{Area_A=Area_B}$
$(100)(40)-\mathrm{Area_B=Area_B}$
$\quad \quad +\mathrm{Area_B \quad +Area_B}$
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$(100)(40)=\mathrm{2Area_B}$
$4000=2(100-2x)(40-2x)$
$4000=2(4000-200x-80x+4x^2)$
$4000=8000-400x-160x+8x^2$
$-4000\quad -40000$
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$0=4000-560x+8x^2$
$0=8x^2-560x+400$
$0=8(x^2-70x+500)$

$\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}=x$
$a=1, b=-70, c=500$

$=\dfrac{+70 \pm \sqrt{70^2-4(1)(500}}{2(1)}$
$=\dfrac{70 \pm \sqrt{2900}}{2}$
$\sqrt{2900}=\sqrt{100}\sqrt{29}=10\sqrt{29}$
$=\dfrac{70 \pm 10\sqrt{29}}{2}$
$=35 \pm 5\sqrt{29}$
$35 +5 \sqrt{29} > 40$ NOT A SOLUTION!
$\boxed{35-5 \sqrt{29} =8.074\mathrm{ft}}$