Factoring Quadratics

Factoring quadratics represents a quadratic expression as the product of its linear factors. A quadratic polynomial is expressed in the form:
                                           

 

ax2 + bx + c = 0

 

Here, a, b, and c represent real numbers.

Solving quadratic equations by factoring is also known as the factorization method.

 

It is used for writing quadratic equations of the form ax² + bx + c = 0 as a product of two linear expressions, typically in the form (x - h)(x - k) k and h are the roots of the equation.

 

A quadratic equation can be factored by splitting the middle term, applying the quadratic formula, and completing the square.

Depending on the structure of the quadratic equation, it can be factored using various methods. These include: 

 

Factoring by splitting the middle term

 

The middle terms are split such that their sum equals the coefficients of x and their product equals the product of x2 and the constant term.

 

• Let's begin with a quadratic equation in the form ax² + bx + c = 0.
   
• We then find two numbers whose sum is b and whose product is equal to a × c.
   
• The middle term is rewritten as split terms
   
• Group the terms into pairs.
   
• Find the GCF from each group.
   
• Factor out the common binomial expression.

 

Example: Factorize 3x² + 11x + 6

• To split the middle term, find numbers having a sum of 11 and a product of  3 × 6 = 18. 9 and 2 are the required numbers.
   
• Rewrite the middle term: 3x² + 9x + 2x + 6
   
• Group the terms (3x² + 9x) + (2x + 6)
   
• Factor 3x(x + 3) + 2(x + 3)
   
• Factor (3x + 2)(x + 3)

Therefore, 3x² + 11x + 6 = (3x + 2)(x + 3)

 

Factoring using the formula

 

We use this method when x² has the coefficient 1 and the quadratic is expressed as (x + a)(x + b).

 

Steps:

 

Write the quadratic in its standard form, i.e.,  x² + bx + c.

Find two numbers, whose sum is b and product is c.

Rewrite the quadratic using the factors.

Example: Factorize x² + 7x + 12

 

• We use the numbers 3 and 4, because,  3 + 4 = 7 and 3 × 4 = 12
   
• Factor: (x + 3)(x + 4)

 

Thus, x² + 7x + 12 = (x + 3)(x + 4)

 

Factoring using the quadratic formula

 

Also known as Shridharacharya’s formula, this is a universal method for solving and factorizing quadratic equations.

 

The formula is: x = -b  b2- 4ac2a     

 

Steps:

• The equation should be written in standard form, that is, ax² + bx + c = 0
   
• Then the quadratic formula is used for finding roots x₁ and x₂.
   
• This gives us the factored form a(x - x₁)(x - x₂)

Example: Factorize 2x² - 5x - 3

 

Use the quadratic formula:
 x = - (-5)   (-5 )2 - 4(2)(-3)2(2)  =   5   25 + 244   = 5   454     = 5   74

 

Roots: x₁ = 3, x₂ = -½

Rewrite: 2(x - 3)(x + ½)

 

Thus, 2x² - 5x - 3 = 2(x - 3)(x + ½)

 

Now we will simplify it further.
(x - 3) (x + 12) = x2 + 12x - 3x - 32 = x2 - 52x - 32

 

Multiplying everything by 2, we get
2 (x2 - 52x - 32) = 2x2 - 5x - 3

 

So the simplified expression is 2x2 - 5x - 3

 

Factoring Using Algebraic Identities

 

This method uses well-known algebraic identities to factorize special forms of quadratic expressions quickly.

 

Common identities used:

 

• (a + b)² = a² + 2ab + b²
   
• (a - b)² = a² - 2ab + b²
   
• a² - b² = (a + b)(a - b)

Steps:

 

• Recognize the algebraic identity that can be applied to the given equation.
   
• Apply the appropriate identity to factorize.

Example: Factorize 9x² - 16

 

This equation is a difference of squares: (3x)² - (4)²

 

So, we apply the identity: a² - b² = (a + b)(a - b),

 

This gives us, 9x² - 16 = (3x + 4)(3x - 4)

To find the roots and factors of a quadratic equation, we use the formula for factoring quadratics. So, we can equate ax2 + bx + c = 0. This is a quadratic equation and;

• a is the coefficient of x2
   
• b is the coefficient of x, and
   
• c is the constant term

 

Then, the value of x can be found using the formula: x = -b  b2 - 4ac2a

 

Let's consider a quadratic equation, 2x2 - 4x + 6 = 0. We can find the value of x using the formula:

x = -b  b2 - 4ac2a

a = 2

b = -4

c = 6

 

Substitute the values of a, b, and c

 

 x = -(-4)  (-4)2 - 4(2)(6)2(2)

 x = 4  16 - 484

 x = 4  16 - 484

 x = 4  -324

 -32  = -1162 = 42i

 

There are two solutions

 

x = 4 + 42i4 

x = 12i 

 

These are not real numbers.

 

So the complex solutions are x = 1 + 2i or x = 1 - 2i

Factoring quadratics helps solve problems in domains involving areas, motion, and optimization scenarios. Some practical applications of factoring quadratics are listed below:

 

Calculating points in time in projectile motion

When an object is in projectile motion, factoring plays a role in determining specific points in time when an object reaches a height and returns.

 

Solving geometry-based problems

Quadratic equations often arise when working with shapes whose area depends on one or more variable dimensions. Factoring allows us to find possible values for those dimensions that meet certain conditions, like fitting within a space or having a fixed total area.

 

Calculating break-even points

Profit or cost functions can sometimes be modeled using quadratic equations. Factoring helps identify where profits become zero, known as break-even points.

 

Solving structural design problems

Engineers and architects use quadratic equations to model the shapes and forces in structures like bridges or arches. Factoring helps find key points like where a structure touches the ground, reaches maximum height, or needs reinforcement.

 

Programming realistic motion in game design

In digital environments, objects often move in curved (parabolic) paths, especially when simulating gravity. Factoring is used to calculate when and where objects will collide, land, or reach a certain point, ensuring the motion looks realistic.