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Solving quadratic equations by factoring is one of the methods to find a solution for the roots of the given polynomial equation. It is also one of the easiest ways to solve quadratic equations. A polynomial equation is an algebraic equation with different terms like variables with varying coefficients, powers, and constants. There happens to be only one variable in a polynomial equation, but the variable has different coefficients within the equation. The equation contains different operations like addition, subtraction, multiplication, and division.
Quadratic Equation
A quadratic polynomial equation has the highest power on the variable as 2.
Thus, a quadratic polynomial equation results in terms of the square root of the polynomial.
The equation has one variable (say, ‘x’) accompanied by various coefficients and powers (say, 3x²+ 13x).
A quadratic equation is always in the format ax² + bx + c.
The term ‘a’ in a quadratic equation is always ‘non-zero’.
Solving Quadratic Equation by Factoring
A quadratic polynomial equation can be solved by one of the three basic methods.
Factoring
Using the quadratic formula
Completing the square
Here, we will be discussing the factoring method. The factors of a number are a pair of smaller numbers that give the larger value of multiplying with each other. For example:
The factors of 15 are 5 and 3.
The factors of 16 are 8, 2 and 4.
Procedure of Solving
Any given quadratic equation can be solved by the factoring method. Consider an equation 3x² + 13x + 4. It is a quadratic polynomial equation.
Split the middle term
Split the middle term (13x) into two terms in such a way that
The sum of the two numbers gives the middle term value (13)
The product of the two numbers is equal to the product of the coefficient of x² is 3 and the constant 4, that is 12×2.
The middle term(13x) is split up into 2 terms (12x and 1x)
3x² + 13x + 4
3x² + 12x + 1x + 4
Take common terms out
Group two terms that have common terms. This will result in the terms inside the parentheses being the same.
3x²+12x+1x+4
In the first and second terms (3x² and 12x, respectively), 3x is common.
In the third and fourth terms (1x and 4 respectively), 1 is the common term.
Take the common terms out
3x (x+4) + 1 (x+4)
Again, from the resulting equation, take a common term out, grouping the previously taken common terms.
3x (x+4) + 1 (x+4)
= (x+4) × (3x+1)
The common term(x+4) has been taken out from the equation.
So, the earlier taken common terms (3x and 1) are grouped.
Equate the terms to find the root
Equate the terms within both the parentheses, separately, to zero.
Taking x+4 = 0
x = -4
Taking 3x+1 = 0,
3x = -1
x = -1/3
x values are the factors
x = -4 or -1/3
Solved Examples of Quadratic Equations by Factoring Method
A few solved examples of quadratic equations by the factoring method are given below:
Solve the quadratic equation 3x² – 2x – 8 by the factoring method.
3x² – 2x – 8
= 3x² + 4x – 6x – 8
= x (3x + 4) – 2 (3x + 4)
= (3x + 4) (x – 2)
Equating 3x + 4,
3x + 4 = 0
3x = -4
x = -4/3
Equating x – 2,
x – 2 = 0
x = 2
Answer: x = -4/3 or 2.
Solve the quadratic equation 6x² + 11x – 35 by the factoring method.
6x² + 11x – 35
= 6x² + 21x – 10x – 35
= 3x (2x + 7) – 5 (2x + 7)
= (2x + 7) (3x – 5)
Equating 2x + 7,
2x + 7 = 0
2x = -7
x = -7/2
Equating 3x – 5,
3x – 5 = 0
3x = 5
x = 5/3
Answer: x = -7/2 or 5/3.
Solve the quadratic equation 2x² – 3x – 5 by the factoring method.
2x² – 3x – 5
= 2x² + 2x – 5x – 5
= 2x (x + 1) – 5 (x + 1)
= (x + 1) (2x – 5)
Equating x + 1,
x + 1 = 0
x = -1
Equating 2x – 5 = 0,
2x – 5 = 0
2x = 5
x = 5/2
Answer: x = -1, 5/2.
Solve the quadratic equation 2(x² + 1) = 5x by the factoring method.
2(x² + 1) = 5x
Simplifying and rearranging the equation to move the terms to left:
= 2x² + 2 – 5x
= 2x² – 5x + 2
= 2x² – 4x – x + 2
= 2x (x – 2) – 1 (x – 2)
= (x – 2) (2x – 1)
Equating x – 2 = 0,
x = 2
Equating 2x – 1 = 0,
2x = 1
x = 1/2
Answer: x = 2, 1/2.
Solve the quadratic equation x² – 3x – 4 by the factoring method.
x² – 3x – 4
= x² + x – 4x – 4
= x (x + 1) – 4 (x + 1)
= (x + 1) (x – 4)
Equating x + 1,
x = -1
Equating x – 4,
x = 4
Answer: x = -1, 4.
Conclusion
A quadratic equation has the highest exponential as 2. Thus, they are named ‘quadratic’. Solving quadratic equations by the factoring method is only one of the methods to solve a quadratic polynomial equation. There are two more methods to solve a quadratic equation. Those methods are the quadratic formula method and completing the square method.
