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Question

2x2-7x=39

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Solution

Step by step solution : Step 1 : Equation at the end of step 1 : (2x2 - 7x) - 39 = 0 Step 2 : Trying to factor by splitting the middle term

2.1 Factoring 2x2-7x-39

The first term is, 2x2 its coefficient is 2 .
The middle term is, -7x its coefficient is -7 .
The last term, "the constant", is -39

Step-1 : Multiply the coefficient of the first term by the constant 2 • -39 = -78

Step-2 : Find two factors of -78 whose sum equals the coefficient of the middle term, which is -7 .

-78 + 1 = -77
-39 + 2 = -37
-26 + 3 = -23
-13 + 6 = -7 That's it


Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -13 and 6
2x2 - 13x + 6x - 39

Step-4 : Add up the first 2 terms, pulling out like factors :
x • (2x-13)
Add up the last 2 terms, pulling out common factors :
3 • (2x-13)
Step-5 : Add up the four terms of step 4 :
(x+3) • (2x-13)
Which is the desired factorization

Equation at the end of step 2 : (2x - 13) • (x + 3) = 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 : 2x-13 = 0

Add 13 to both sides of the equation :
2x = 13
Divide both sides of the equation by 2:
x = 13/2 = 6.500

Solving a Single Variable Equation :

3.3 Solve : x+3 = 0

Subtract 3 from both sides of the equation :
x = -3


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