5x2 + 5x +1 split the middle term
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Answers
Two solutions were found :
x = -5
x = 1/5 = 0.200
Step by step solution :
Step 1 :
Equation at the end of step 1 :
((0 - 5x2) - 24x) + 5 = 0
Step 2 :
Step 3 :
Pulling out like terms :
3.1 Pull out like factors :
-5x2 - 24x + 5 = -1 • (5x2 + 24x - 5)
Trying to factor by splitting the middle term
3.2 Factoring 5x2 + 24x - 5
The first term is, 5x2 its coefficient is 5 .
The middle term is, +24x its coefficient is 24 .
The last term, "the constant", is -5
Step-1 : Multiply the coefficient of the first term by the constant 5 • -5 = -25
Step-2 : Find two factors of -25 whose sum equals the coefficient of the middle term, which is 24 .
-25 + 1 = -24
-5 + 5 = 0
-1 + 25 = 24 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -1 and 25
5x2 - 1x + 25x - 5
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (5x-1)
Add up the last 2 terms, pulling out common factors :
5 • (5x-1)
Step-5 : Add up the four terms of step 4 :
(x+5) • (5x-1)
Which is the desired factorization
Equation at the end of step 3 :
(1 - 5x) • (x + 5) = 0
Step 4 :
Theory - Roots of a product :
4.1 A product of several terms equals zero.
When a product of two or more terms equals zero, then at least one of the terms must be zero.
We shall now solve each term = 0 separately
In other words, we are going to solve as many equations as there are terms in the product
Any solution of term = 0 solves product = 0 as well.
Solving a Single Variable Equation :
4.2 Solve : -5x+1 = 0
Subtract 1 from both sides of the equation :
-5x = -1
Multiply both sides of the equation by (-1) : 5x = 1
Divide both sides of the equation by 5:
x = 1/5 = 0.200
Solving a Single Variable Equation :
4.3 Solve : x+5 = 0
Subtract 5 from both sides of the equation :
x = -5
Supplement : Solving Quadratic Equation Directly
Solving 5x2+24x-5 = 0 directly
Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula