factorize 5x2-4x-33
Answers
Answer:
Two solutions were found :
x = -11/5 = -2.200
x = 3
Step by step solution :
Step 1 :
Equation at the end of step 1 :
(5x2 - 4x) - 33 = 0
Step 2 :
Trying to factor by splitting the middle term
2.1 Factoring 5x2-4x-33
The first term is, 5x2 its coefficient is 5 .
The middle term is, -4x its coefficient is -4 .
The last term, "the constant", is -33
Step-1 : Multiply the coefficient of the first term by the constant 5 • -33 = -165
Step-2 : Find two factors of -165 whose sum equals the coefficient of the middle term, which is -4 .
-165 + 1 = -164
-55 + 3 = -52
-33 + 5 = -28
-15 + 11 = -4 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -15 and 11
5x2 - 15x + 11x - 33
Step-4 : Add up the first 2 terms, pulling out like factors :
5x • (x-3)
Add up the last 2 terms, pulling out common factors :
11 • (x-3)
Step-5 : Add up the four terms of step 4 :
(5x+11) • (x-3)
Which is the desired factorization
Equation at the end of step 2 :
(x - 3) • (5x + 11) = 0
Step 3 :
Theory - Roots of a product :
3.1 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 :
3.2 Solve : x-3 = 0
Add 3 to both sides of the equation :
x = 3
Solving a Single Variable Equation :
3.3 Solve : 5x+11 = 0
Subtract 11 from both sides of the equation :
5x = -11
Divide both sides of the equation by 5:
x = -11/5 = -2.200
Supplement : Solving Quadratic Equation Directly
Solving 5x2-4x-33 = 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...
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