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How do you factor 2x2-13x-7 = 0?
First let's determine how to multiply 2 binomials
Do you know the FOIL rule? FOIL stands for first, outer, inner, last.
Equations like the one above are the result of multiplying two binomials such as:
(x + 5) (2x - 3) = 0
First tells you to multiply the variable in the first binomial times the variable in the second binomial. x ·2x = 2x2
Outer tells you to multiply the variable in the first binomial times the number in the second binomial. x · -3 = -3x (notice subtract 3 has changed to -3)
Inner tells you to multiply the number in the first binomial times the variable in the second binomial. +5 · 2x = 10x
Last tells you to multiply the number in the first binomial times the number in the second binomial. +5 · -3 = -15 (notice subtract 3 has changed to -3)
Now add all 4 answers.
2x2 + -3x + 10x + -15 = 0
2x2 + ( -3x + 10x) + -15 = 0
2x2 + 7x - 15 = 0 (notice + -15 has changed to subtract 15)
Notice the 2 in 2x2 is the product of the coefficients of the variables (First), so you are looking for factors of 2.
Notice the -15 is the product of the numbers (Last), so you are looking for factors of -15.
Notice the +7x is the Sum of the Outer and Inner answers.
How do you factor 2x2 -13x -7 = 0?
Factors of 2 are (1 · 2), and (2 · 1).
Factors of -7 are (-7· 1) and (1· -7).
The Sum of the Outer and Inner answers must equal -13.
There are 4 possible combinations
- (1 · 2) (-7· 1) (x -7) · (2x +1) = 0
- (1 · 2) (1· -7) (x -1) · (2x +7) = 0
- (1 · 2) (1· -7) (x +1) · (2x -7) = 0
- (1 · 2) (-7· 1) (x -7) · (2x +1) = 0
In which of these equations do the sum of the Outer and Inner answers equal -13.
1. (x - 7) · (2x +1) = 0 Outer = +1, Inner = -14 Sum = -13
First = 2x2 Last = -7
(2x +1) (x -7) = 2x2-13x -7 = 0
If the coefficients of the variables and the numbers have more than one possible set of factors, the process becomes more complex. But keep practicing and you will do well!
After you practice writing the possible combinations, it will become more obvious which combination is correct. You will be able to do this in your head without writing all the possible combinations.
2x2 -x -15 = 0
Factors of 2 are (1 · 2) Factors of -15 are (1 · -15), (-1 · 15), (3 · -5) (-3 · 5)
1. (x + 1) · (2x - 15) = 0
2. (x -15) · (2x + 1) = 0
3. (x +3) · (2x - 5) = 0
4. (x -3) · (2x +5) = 0
5. (x -5) · (2x + 3) = 0
6. (x -5) · (2x - 3) = 0
In which of these equations, do the sum of the Outer and Inner answers must equal -1.
5. (x -3) · (2x +5) = 2x2 -6x + 5x -15 = 2x2 -x -15
Do you see 3, 5 and 2, 2· 3 = 6 -5 = 1. Learn to adjust the signs and you will be a winner.
Improved answer:
Using the quadratic equation formula will make it easier to solve this problem which will have two solutions:
2x2-13x-7 = 0
When factorised:
(2x+1)(x-7) = 0
Therefore: x = -1/2 or x = 7
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