The following steps will be useful to factor a quadratic equation.

Step 1 :

Write the equation in form ax2 + bx + c  =  0.

Step 2 :

If the coefficient of x2 is 1, we have to take the constant term and split it into two factors such that the product of those factors must be equal to the constant term and simplified value must be equal to the middle term.

If the coefficient of x2 is not 1, we have to multiply the constant term along with the coefficient of x2.Split the product into two factors.

Step 3 :

Rewrite the middle with those numbers.

Step 4 :

Factor the first two and last two separately.

Step 5 :

Equate the linear factors to zero and solve for x.

## Solved Examples

Example 1 :

Solve x2 + 17x + 60  =  0

Solution :

x2 + 17x + 60  =  0

60  =  12 ⋅ 5 and 17  =  12 + 5

Factors of 60 are 12 and 5. By multiplying 12 and 5, we get 60 and simplifying 12 and 5, we get 17.

x2 + 12x + 5x + 60  =  0

x2 + 12x + 5x + 60  =  0

x(x + 12) + 5(x + 12)  =  0

(x + 5) (x + 12)  =  0

 x + 5  =  0x  =  -5 x + 12  =  0x  =  -12

So, the solution is {-5, -12}.

Example 2 :

Solve x2 - 5x - 36  =  0

Solution :

x2 - 5x - 36  =  0

-36  =  -9 ⋅ 4 and -5  =  -9 + 4

Factors of -36 are -9 and 4. By multiplying -9 and 4, we get -36 and simplifying -9 and 4, we get -5.

x2 - 9x + 4x - 36  =  0

x2 - 9x + 4x - 36  =  0

x(x - 9) + 4(x - 9)  =  0

(x - 9) (x + 4)  =  0

 x - 9  =  0x  =  9 x + 4  =  0x  =  -4

So, the solution is {-4, 9}.

Example 3 :

Solve x2 - 14x + 48  =  0

Solution :

x2 - 14x + 48  =  0

48  =  -8 ⋅ (-6) and -14  =  -8 - 6

Factors of 48 are -8 and -6. By multiplying -8 and -6, we get 48 and simplifying -8 and -6, we get -14.

x2 - 8x - 6x + 48  =  0

x2 - 8x - 6x + 48  =  0

x(x - 8) - 6(x - 8)  =  0

(x - 8) (x - 6)  =  0

 x - 8  =  0x  =  8 x - 6  =  0x  =  6

So, the solution is {6, 8}.

Example 4 :

Solve x2 + 2x - 24  =  0

Solution :

x2 + 2x - 24  =  0

-24  =  6 ⋅ (-4) and 2  =  6 - 4

Factors of -24 are 6 and -4. By multiplying 6 and -4, we get -24 and simplifying 6 and -4, we get 2.

x2 + 6x - 4x - 24  =  0

x2 + 6x - 4x - 24  =  0

x(x + 6) - 4(x + 6)  =  0

(x + 6) (x - 4)  =  0

 x + 6  =  0x  =  -6 x - 4  =  0x  =  4

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