Answer:
20.25
Step-by-step explanation:
8 x 1.75 = 14
5 x 1.25 = 6.25
![g(x) = 2(x-2)^2-5 \text{ is equivalent to } 2x^2-8x+3](https://tex.z-dn.net/?f=g%28x%29%20%3D%202%28x-2%29%5E2-5%20%5Ctext%7B%20is%20equivalent%20to%20%7D%202x%5E2-8x%2B3)
<em><u>Solution:</u></em>
Given expression is:
![f(x) = 2x^2-8x+3](https://tex.z-dn.net/?f=f%28x%29%20%3D%202x%5E2-8x%2B3)
We have to find the equivalent expression
![\mathrm{Write}\:2x^2-8x+3\:\mathrm{in\:the\:form:\:\:}x^2+2ax+a^2](https://tex.z-dn.net/?f=%5Cmathrm%7BWrite%7D%5C%3A2x%5E2-8x%2B3%5C%3A%5Cmathrm%7Bin%5C%3Athe%5C%3Aform%3A%5C%3A%5C%3A%7Dx%5E2%2B2ax%2Ba%5E2)
![f(x) = 2x^2-8x+3](https://tex.z-dn.net/?f=f%28x%29%20%3D%202x%5E2-8x%2B3)
![\mathrm{Factor\:out\:}2](https://tex.z-dn.net/?f=%5Cmathrm%7BFactor%5C%3Aout%5C%3A%7D2)
![2\left(x^2-4x+\frac{3}{2}\right)](https://tex.z-dn.net/?f=2%5Cleft%28x%5E2-4x%2B%5Cfrac%7B3%7D%7B2%7D%5Cright%29)
![\mathrm{Add\:and\:subtract}\:\left(-2\right)^2\:](https://tex.z-dn.net/?f=%5Cmathrm%7BAdd%5C%3Aand%5C%3Asubtract%7D%5C%3A%5Cleft%28-2%5Cright%29%5E2%5C%3A)
![2\left(x^2-4x+\frac{3}{2}+\left(-2\right)^2-\left(-2\right)^2\right)](https://tex.z-dn.net/?f=2%5Cleft%28x%5E2-4x%2B%5Cfrac%7B3%7D%7B2%7D%2B%5Cleft%28-2%5Cright%29%5E2-%5Cleft%28-2%5Cright%29%5E2%5Cright%29)
Simplify the above equation
![2(x^2-4x + \frac{3}{2} + 4 -4)\\\\2(x^2-4x +4 + \frac{3}{2} - 4)\\\\2(x^2-4x +4 -\frac{5}{2}) ------- eqn 1](https://tex.z-dn.net/?f=2%28x%5E2-4x%20%2B%20%5Cfrac%7B3%7D%7B2%7D%20%2B%204%20-4%29%5C%5C%5C%5C2%28x%5E2-4x%20%2B4%20%2B%20%5Cfrac%7B3%7D%7B2%7D%20-%204%29%5C%5C%5C%5C2%28x%5E2-4x%20%2B4%20-%5Cfrac%7B5%7D%7B2%7D%29%20-------%20eqn%201)
Using the algebraic identity,
![x^2+2ax+a^2=\left(x+a\right)^2](https://tex.z-dn.net/?f=x%5E2%2B2ax%2Ba%5E2%3D%5Cleft%28x%2Ba%5Cright%29%5E2)
Therefore,
![x^2-4x+4=\left(x-2\right)^2](https://tex.z-dn.net/?f=x%5E2-4x%2B4%3D%5Cleft%28x-2%5Cright%29%5E2)
Substitute the above in eqn 1
![2((x-2)^2-\frac{5}{2})](https://tex.z-dn.net/?f=2%28%28x-2%29%5E2-%5Cfrac%7B5%7D%7B2%7D%29)
Simplify the above equation by multiplying 2 with terms inside the bracket
![2(x-2)^2 -5](https://tex.z-dn.net/?f=2%28x-2%29%5E2%20-5)
Thus the equivalent expression is ![g(x) = 2(x-2)^2-5](https://tex.z-dn.net/?f=g%28x%29%20%3D%202%28x-2%29%5E2-5)
Answer:
![m\angle QSN=65^\circ](https://tex.z-dn.net/?f=m%5Cangle%20QSN%3D65%5E%5Ccirc)
Step-by-step explanation:
In the given figure, PQRS is a rhombus and SRM is an equilateral triangle.
We are also given that SN⊥RM and that ∠PRS = 55°.
And we want to find the measure of ∠QSN.
Remember that since PQRS is a rhombus, the angles formed by its diagonals are right angles. Let the intersection point of the diagonals be K. Therefore:
![m\angle RKS=90^\circ](https://tex.z-dn.net/?f=m%5Cangle%20RKS%3D90%5E%5Ccirc)
Now, RKS is also a triangle. The interior angles of all triangles must be 180. Thus:
![m\angle RKS+m\angle KSR+m\angle SRK=180](https://tex.z-dn.net/?f=m%5Cangle%20RKS%2Bm%5Cangle%20KSR%2Bm%5Cangle%20SRK%3D180)
Substitute in known values:
![90+55+m\angle KSR=180](https://tex.z-dn.net/?f=90%2B55%2Bm%5Cangle%20KSR%3D180)
Solve for ∠KSR:
![m\angle KSR+145=180\Rightarrow m\angle KSR=35^\circ](https://tex.z-dn.net/?f=m%5Cangle%20KSR%2B145%3D180%5CRightarrow%20m%5Cangle%20KSR%3D35%5E%5Ccirc)
Since SRM is an equilateral triangle, this means that:
![m\angle SRM=m\angle RMS=m\angle MSR=60^\circ](https://tex.z-dn.net/?f=m%5Cangle%20SRM%3Dm%5Cangle%20RMS%3Dm%5Cangle%20MSR%3D60%5E%5Ccirc)
Note that RNS is also a triangle. Therefore:
![m\angle SRM+m\angle RNS+m\angle NSR=180](https://tex.z-dn.net/?f=m%5Cangle%20SRM%2Bm%5Cangle%20RNS%2Bm%5Cangle%20NSR%3D180)
Substitute in known values:
![60+90+m\angle NSR=180](https://tex.z-dn.net/?f=60%2B90%2Bm%5Cangle%20NSR%3D180)
So:
![m\angle NSR+150=180\Rightarrow m\angle NSR=30^\circ](https://tex.z-dn.net/?f=m%5Cangle%20NSR%2B150%3D180%5CRightarrow%20m%5Cangle%20NSR%3D30%5E%5Ccirc)
∠QSN is the addition of the two angles:
![m\angle QSN=m\angle KSR+m\angle NSR](https://tex.z-dn.net/?f=m%5Cangle%20QSN%3Dm%5Cangle%20KSR%2Bm%5Cangle%20NSR)
Therefore:
![m\angle QSN=35+30=65^\circ](https://tex.z-dn.net/?f=m%5Cangle%20QSN%3D35%2B30%3D65%5E%5Ccirc)
I would start at making it more easy by asking my dad or mom for help