It is a very simple answer but it works!
Consider right triangle ΔABC with legs AC and BC and hypotenuse AB. Draw the altitude CD.
1. Theorem: The length of each leg of a right triangle is the geometric mean of the length of the hypotenuse and the length of the segment of the hypotenuse adjacent to that leg.
According to this theorem,
![BC^2=BD\cdot AB.](https://tex.z-dn.net/?f=BC%5E2%3DBD%5Ccdot%20AB.)
Let BC=x cm, then AD=BC=x cm and BD=AB-AD=3-x cm. Then
![x^2=(3-x)\cdot 3,\\ \\x^2=9-3x,\\ \\x^2+3x-9=0,\\ \\D=3^2-4\cdot (-9)=9+36=45,\\ \\\sqrt{D}=\sqrt{45}=3\sqrt{5},\\ \\x_1=\dfrac{-3-3\sqrt{5} }{2}0.](https://tex.z-dn.net/?f=x%5E2%3D%283-x%29%5Ccdot%203%2C%5C%5C%20%5C%5Cx%5E2%3D9-3x%2C%5C%5C%20%5C%5Cx%5E2%2B3x-9%3D0%2C%5C%5C%20%5C%5CD%3D3%5E2-4%5Ccdot%20%28-9%29%3D9%2B36%3D45%2C%5C%5C%20%5C%5C%5Csqrt%7BD%7D%3D%5Csqrt%7B45%7D%3D3%5Csqrt%7B5%7D%2C%5C%5C%20%5C%5Cx_1%3D%5Cdfrac%7B-3-3%5Csqrt%7B5%7D%20%7D%7B2%7D%3C0%2C%5C%20x_2%3D%5Cdfrac%7B-3%2B3%5Csqrt%7B5%7D%20%7D%7B2%7D%3E0.)
Take positive value x. You get
![AD=BC=\dfrac{-3+3\sqrt{5} }{2}\ cm.](https://tex.z-dn.net/?f=AD%3DBC%3D%5Cdfrac%7B-3%2B3%5Csqrt%7B5%7D%20%7D%7B2%7D%5C%20cm.)
2. According to the previous theorem,
![AC^2=AD\cdot AB.](https://tex.z-dn.net/?f=AC%5E2%3DAD%5Ccdot%20AB.)
Then
![AC^2=\dfrac{-3+3\sqrt{5} }{2}\cdot 3=\dfrac{-9+9\sqrt{5} }{2},\\ \\AC=\sqrt{\dfrac{-9+9\sqrt{5} }{2}}\ cm.](https://tex.z-dn.net/?f=AC%5E2%3D%5Cdfrac%7B-3%2B3%5Csqrt%7B5%7D%20%7D%7B2%7D%5Ccdot%203%3D%5Cdfrac%7B-9%2B9%5Csqrt%7B5%7D%20%7D%7B2%7D%2C%5C%5C%20%5C%5CAC%3D%5Csqrt%7B%5Cdfrac%7B-9%2B9%5Csqrt%7B5%7D%20%7D%7B2%7D%7D%5C%20cm.)
Answer: ![AC=\sqrt{\dfrac{-9+9\sqrt{5} }{2}}\ cm.](https://tex.z-dn.net/?f=AC%3D%5Csqrt%7B%5Cdfrac%7B-9%2B9%5Csqrt%7B5%7D%20%7D%7B2%7D%7D%5C%20cm.)
This solution doesn't need CD=2 cm. Note that if AB=3cm and CD=2cm, then
![CD^2=AD\cdot DB,\\ \\2^2=AD\cdot (3-AD),\\ \\AD^2-3AD+4=0,\\ \\D](https://tex.z-dn.net/?f=CD%5E2%3DAD%5Ccdot%20DB%2C%5C%5C%20%5C%5C2%5E2%3DAD%5Ccdot%20%283-AD%29%2C%5C%5C%20%5C%5CAD%5E2-3AD%2B4%3D0%2C%5C%5C%20%5C%5CD%3C0)
This means that you cannot find solutions of this equation. Then CD≠2 cm.
Sophie has factored the term completely while Carlos has not factored the term completely.
Step-by-step explanation:
We need to factor the trinomial:
![-4x^2-10x + 6](https://tex.z-dn.net/?f=-4x%5E2-10x%20%2B%206)
Factoring:
![-4x^2-10x + 6\\Taking\,\,-2\,\,common:\\-2(2x^2+5x-3)\\Now\,\,factoring:\\-2(2x^2+6x-x-3)\\-2(2x(x+3)-1(x+3))\\-2(2x-1)(x+3)](https://tex.z-dn.net/?f=-4x%5E2-10x%20%2B%206%5C%5CTaking%5C%2C%5C%2C-2%5C%2C%5C%2Ccommon%3A%5C%5C-2%282x%5E2%2B5x-3%29%5C%5CNow%5C%2C%5C%2Cfactoring%3A%5C%5C-2%282x%5E2%2B6x-x-3%29%5C%5C-2%282x%28x%2B3%29-1%28x%2B3%29%29%5C%5C-2%282x-1%29%28x%2B3%29)
So, the factored form of
is ![-2(2x-1)(x+3)](https://tex.z-dn.net/?f=-2%282x-1%29%28x%2B3%29)
So, Sophie has factored the term completely. The terms are factored when we cannot further simplify it. Since Carlos factored (x+3)(-4x+2) 2 can be taken common from the term (-4x+2) so, it is not factored completely.
Keywords: Factoring the terms
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