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Factoring Quadratic Equations when a = 1

A quadratic equation is an equation that contains a squared variable as its highest power on any variable. The general form of a quadratic equation is:

ax2 + bx + c = 0


Where a, b, and c are constants and a ≠ 0. In other words there must be a x2 term.

Some examples are:

x2 + 3x - 3 = 0
4x2 + 9 = 0 (Where b = 0)
x2 + 5x = 0 (where c = 0)

One way to solve a quadratic equation is by factoring the trinomial.

This part will focus on factoring a quadratic when a, the x2-coefficient, is 1.

ax2 + bx + c = 0 → (x + r) (x + s)




Let's solve the following equation by factoring the trinomial:

x2 + 5x - 14 = 0

Step 1: Write the equation in the general form

ax2 + bx + c = 0.


This equation is already in the proper form where a = 1, b = 5 and c = -14.

1x2 + 5x - 14 = 0

Step 2: Determine the factor pair of c that will add to give b.


2.1: List the factor pairs of c.


First ask yourself what are the factors pairs of c, ignoring the negative sign for now.


2.2: Determine the signs of the factors.


If c is positive then both factors will be positive or both factors will be negative.


If c is negative then one factor will be positive and the other negative. Now create factor pairs


2.3: Determine the factor pair that will add to give b.


If both c and b are positive, both factors will be positive.


If both c and b are negative, the larger factor will be negative and the smaller will be positive.


If c is positive and b is negative, both factors will be negative.


If c is negative and b is positive, the larger factor will be positive and the smaller will be negative.




(1, 14); (2, 7)




2.2: c = -14, a negative number, therefore one factor is negative and the other is positive.






2.3: b = 5, a positive number, therefore the larger factor will be positive and the smaller will be negative.


( 14,−1 ): 14+( −1 )=13≠ b


This pair does not work.


( 7,−2 ): 7+( −2 )=5=b


This pair works!!!


(7, -2)

Step 3:

Create two sets of parentheses each containing a x and one of the factors.

(x + 7)(x - 2) = 0

Now that the equation has been factored, solve for x.

Step 4: Set each factor to zero and solve for x.

(x + 7) = 0, or (x - 2) = 0


x = -7, or x = 2

Example 1:      8x + 15 = x2

Step 1: Write the equation in the general form

ax2 + bx + c = 0.


Where a = 1, b = 8 and c = 15.

x2 + 8x + 15 = 0

Step 2: Determine the factor pair of c that will add to give b.


2.1: List the factor pairs of c.


First ask yourself what are the factors pairs of c, ignoring the negative sign for now.


2.2: Determine the signs of the factors.


If c is positive then both factors will be positive or both factors will be negative.


If c is negative then one factor will be positive and the other negative. Now create factor pairs


2.3: Determine the factor pair that will add to give b.


If both c and b are positive, both factors will be positive.


If both c and b are negative, the larger factor will be negative and the smaller will be positive.


If c is positive and b is negative, both factors will be negative.


If c is negative and b is positive, the larger factor will be positive and the smaller will be negative.




2.1: Factors pairs of 15:

(1, 15);(3,5)




2.2: c = 15, a positive number, therefore both factors will be positive or both factors will be negative.






2.3: b = 8, a positive number, therefore the both factors will be positive.


( 15,1 ): 15+1=16≠ b


This pair does not work.


( 3,5 ): 3+5=8=b


This pair works!!!


(3, 5)

Step 3:

Create two sets of parentheses each containing a x and one of the factors.

(x + 3)(x + 5)

Now that the equation has been factored, solve for x.

Step 4: Set each factor to zero and solve for x.

(x + 3) = 0, or (x + 5) = 0


x= -3, or x = -5

Example 2:      -24 + x2 = -10x

Step 1: Write the equation in the general form

ax2 + bx + c = 0.


Where a = 1, b = 10 and c = -24.

x2 + 10x - 24 = 0

Step 2: Determine the factor pair of c that will add to give b.


2.1: List the factor pairs of c.


First ask yourself what are the factors pairs of c, ignoring the negative sign for now.


2.2: Determine the signs of the factors.


If c is positive then both factors will be positive or both factors will be negative.


If c is negative then one factor will be positive and the other negative. Now create factor pairs


2.3: Determine the factor pair that will add to give b.


If both c and b are positive, both factors will be positive.


If both c and b are negative, the larger factor will be negative and the smaller will be positive.


If c is positive and b is negative, both factors will be negative.


If c is negative and b is positive, the larger factor will be positive and the smaller will be negative.




2.1: Factors pairs of 24:


(1, 24);(2, 12);(3, 8);(6, 4)



2.2: c = -24, a negative number, therefore one factor is negative and the other is positive.





2.3: b = 10, a positive number, therefore the larger factor will be positive and the smaller factor negative.


( −1,24 ): −1+24=23≠ b


( −3,8 ): −3+8=5≠ b


( −4,6 ): −4+6=2≠ b


These pairs do not work.


( −2,12 ): −2+12=10= b


This pair works!!!


(-2, 12)

Step 3:

Create two sets of parentheses each containing a x and one of the factors.

(x - 2)(x + 12)

Now that the equation has been factored, solve for x.

Step 4: Set each factor to zero and solve for x.

(x - 2) = 0, or (x + 12) = 0


x = 2, or x = -12

Factoring quadratics is generally the easier method for solving quadratic equations. However, not all quadratic equations can be factored evenly. In these cases it is usually better to solve by completing the square or using the quadratic formula.




Related Links:
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Factoring Quadratic Equations when a ≠ 1
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