2x2-7x=39
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
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
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.
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
3.3 Solve : x+3 = 0
Subtract 3 from both sides of the equation :
x = -3
