Staircase

On a staircase 3.6 meters high, the number of steps would increase by 3 if the height of one step decreased by 4 cm. How high are the stairs?

Correct answer:

x =  24 cm

Step-by-step explanation:

$$\begin{gathered} h=3.6\text{ m}\to \text{ cm}=3.6\cdot 100\text{ cm}=360\text{ cm } \\ h=n\cdot x \\ h=(n+3)(x-4) \\ h=(h\mathrm{/}x+3)\cdot (x-4) \\ h\ast x=(h+3\ast x)\ast (x-4) \\ 360\cdot x=(360+3\cdot x)\cdot (x-4) \\ -3x^2+12x+1440=0 \\ 3x^2-12x-1440=0 \\ a=3;b=-12;c=-1440 \\ D=b^2-4ac=12^2-4\cdot 3\cdot (-1440)=17424 \\ D>0 \\ {x}_{1,2}={\displaystyle \frac{-b\pm \sqrt{D}}{2a}}={\displaystyle \frac{12\pm \sqrt{17424}}{6}} \\ {x}_{1,2}={\displaystyle \frac{12\pm 132}{6}} \\ {x}_{1,2}=2\pm 22 \\ {x}_1=24 \\ {x}_2=-20 \\ \text{ Factored form of the equation: } \\ 3(x-24)(x+20)=0 \\ x=x_1=24\text{ cm } \\ n=h\mathrm{/}x=360\mathrm{/}24=15 \\ \text{ Verifying Solution: } \\ {h}_1=n\cdot x=15\cdot 24=360\text{ cm } \\ {h}_2=(n+3)\cdot (x-4)=(15+3)\cdot (24-4)=360\text{ cm} \end{gathered}$$

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