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Solve x2+5x50=0.

Answer
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Hint: In order to determine the factors of the above quadratic equation use the Splitting up the middle term method or also called mid-term factorization method. In which the middle term would be splitted into two parts and similar values will be taken common and equated to zero to obtain the roots.

Complete step by step solution:
Since, the highest degree in the given polynomial is two, so it’s a quadratic equation.
Given a quadratic equation x2+5x50 ,let it be f(x)
f(x)=x2+5x50
Comparing the equation with the standard Quadratic equation ax2+bx+c
a becomes 1
b becomes 5
And c becomes 50
To find the quadratic factorization we’ll use splitting up the middle term method
So first calculate the product of coefficient of x2 and the constant term which comes to be
=1×(50)=(50)
Now second Step is to the find the 2 factors of the number 50 such that whether, addition or subtraction of those numbers is equal to the middle term or coefficient of x and the product of those factors results in the value of constant.
So, if we factorize 50 , the answer comes to be 10 and 5 as 105=5 that is the middle term. and 10×(5)=(50) which is perfectly equal to the constant value.
Now writing the middle term sum of the factors obtained, so equation f(x) becomes
f(x)=x2+10x5x50=0
Now taking common from the first 2 terms and last 2 terms
f(x)=x(x+10)5(x+10)=0
Finding the common binomial parenthesis, the equation becomes
f(x)=(x5)(x+10)=0
Hence, we have successfully factorized our quadratic equation.
Therefore, the factors are (x5) and (x+10).
Equating the factors with zero to get the roots:
 (x5)=0x=5
And,
 (x+10)=0x=(10)

Therefore, the roots of the quadratic equation x2+5x50=0 are: x=5 and x=(10).

Note:
Alternative Approach:
You can also alternatively use a direct method which uses Quadratic Formula to find both roots of a quadratic equation as
x1=b+b24ac2a and x2=bb24ac2a
x1, x2 are root to quadratic equation ax2+bx+c
For the equation given above x2+5x50=0 , comparing the equation with the standard Quadratic equation ax2+bx+c
a becomes 1
b becomes 5
And c becomes 50
Substituting these values in x1=b+b24ac2a and x2=bb24ac2a , and we get:
For first root:
x1=b+b24ac2ax1=5+(5)24(1)(50)2(1)x1=5+25+2002x1=5+2252

Since, we know that 225=15 , so opening the parenthesis above:
x1=5+2252x1=5+152=102=5
Therefore, the first root is x1=5.

Similarly, for second root:
x2=bb24ac2ax2=5(5)24(1)(50)2(1)x2=525+2002x2=52252

Since, we know that 225=15 , so opening the parenthesis above:
x2=52252x2=5152=202=(10)
Therefore, the first root is x2=(10).
Hence, the factors will be (x5)and(x+10).

1. Mid-term factorization and Quadratic formulas are always applied on Quadratic Equations.
2. Any one of the above methods can be used to find the roots of the quadratic equation.
3. We can leave the values till factors until it’s given to find the roots.
4. Don’t forget to compare the given quadratic equation with the standard one every time.