How to Compute the Bar Designation Using the Quadratic Equation
Factoring Quadratics
Factoring quadratics is a method of expressing the polynomial as a product of its linear factors. It is a process that allows us to simplify quadratic expressions, find their roots and solve equations. A quadratic polynomial is of the form ax2 + bx + c, where a, b, c are real numbers. Factoring quadratics is a method that helps us to find the zeros of the quadratic equation ax2 + bx + c = 0.
In this mini-lesson, let us learn about the fascinating concept of factoring quadratics, the formula for factorization of quadratic equations along some solved examples for a better understanding.
| 1. | What is Factoring Quadratics? |
| 2. | Methods of Factoring Quadratics |
| 3. | Identities for Factoring Quadratics |
| 4. | Formula for Factoring Quadratics |
| 5. | FAQs on Factoring Quadratics |
What Is Factoring Quadratics?
Factoring quadratics is a method of expressing the quadratic equation ax2 + bx + c = 0 as a product of its linear factors as (x - k)(x - h), where h, k are the roots of the quadratic equation ax2 + bx + c = 0. This method is also is called the method of factorization of quadratic equations. Factorization of quadratic equations can be done using different methods such as splitting the middle term, using the quadratic formula, completing the squares, etc.
Factoring Quadratics Meaning
The factor theorem relates the linear factors and the zeros of any polynomial. Every quadratic equation has two roots, say \(\alpha\) and \(\beta\). They are the zeros of the quadratic equation. Consider a quadratic equation f(x) = 0, where f(x) is a polynomial of degree 2. Suppose that x = \(\alpha\) is one root of this equation. This means that x = \(\alpha\) is a zero of the quadratic expression f(x). Thus, (x - \(\alpha\)) should be a factor of f(x).
Similarly, if x = \(\beta\) is the second root of f(x) = 0, then x = \(\beta\) is a zero of f(x). Thus, (x - \(\beta\)) should be a factor of f(x). Hence, factoring quadratics is a method of expressing the quadratic equations as a product of its linear factors, that is, f(x) = (x - \(\alpha\))(x - \(\beta\)). Let us go through some examples of factoring quadratics:
Examples of Factoring Quadratics
1. Consider the quadratic equation x2 + 5x + 6 = 0
-3 and -2 are the roots of the equation. Verify by substituting the roots in the given equation and check if the value equals 0.
Factor 1: (x + 3)
LHS = x2 + 5x + 6 = (-3)2 + 5 × -3 + 6 = 9 -15 + 6 = 0 = RHS
Factor 2: (x + 2)
LHS = x2 + 5x + 6 = (-2)2 + 5 × -2 + 6 = 4 -10 + 6 = 0 = RHS
Thus the equation has 2 factors (x + 3) and (x + 2)
2. Consider x2 - 9 = 0
3 and -3 are the two roots of the equation. Verify by substituting the roots in the given equation and check if the value equals 0.
32 - 9 = 9 - 9 = 0
(-3)2 - 9 = 9 - 9 = 0
Thus the equation has 2 factors (x+3) and (x-3)
Methods of Factoring Quadratics
Factoring quadratics gives us the roots of the quadratic equation. There are different methods that can be used for factoring quadratic equations. Factoring quadratics is done in 4 ways:
- Factoring out the GCD
- Splitting the middle term
- Using Algebraic Identities (Completing the Squares)
- Using Quadratic formula
Factoring Quadratics by Taking Out The GCD
Factoring quadratics can be done by finding the common numeric factor and the algebraic factors shared by the terms in the quadratic equation and then take them out. Let us solve an example to understand the factoring quadratic equations by taking the GCD out.
Consider this quadratic equation: 3x2 + 6x = 0
- The numerical factor is 3 (coefficient of x2) in both terms.
- The algebraic common factor is x in both terms.
- The common factors are 3 and x. Hence we take them out.
- Thus 3x2 + 6x = 0 is factorized as 3x(x + 2) = 0
Splitting the Middle Term for Factoring Quadratics
- The sum of the roots of the quadratic equation ax2 + bx + c = 0 is given by \(\alpha + \beta\) = -b/a
- The product of the roots in the quadratic equation ax2 + bx + c = 0 is given by \(\alpha\beta\) = c/a
We split the middle term b of the quadratic equation ax2 + bx + c = 0 when we try to factorize quadratic equations. We determine the factor pairs of the product of a and c such that their sum is equal to b.
Example: f(x) = x2 + 8x + 12
Split the middle term 8x in such a way that the factors of the product of 1 and 12 add up to make 8. Factor pairs of 12 are (1, 12), (2, 6), (3, 4). Now, we can see that the factor pair (2, 6) satisfies our purpose as the sum of 6 and 2 is 8 and the product is 12. Hence, we split the middle term and write the quadratic equation as:
x2 + 8x + 12 = 0
⇒ x2 + 6x + 2x + 12 = 0
Now, club the terms in pairs as:
(x2 + 6x) + (2x + 12) = 0
⇒ x(x + 6) + 2(x + 6) = 0
Taking the common factor (x + 6) out, we have
(x + 2) (x + 6) = 0
Thus, (x + 2) and (x + 6) are the factors of x2 + 8x + 12 = 0
Identities for Factoring Quadratics
The process of factoring quadratics can be done by completing the squares which require the use of algebraic identities. The main algebraic identities which are used for completing the squares are:
- (a + b)2 = a2 + 2ab + b2
- (a - b)2 = a2 - 2ab + b2
Steps to factorize quadratic equation ax2 + bx + c = 0 using completeing the squares method are:
- Step 1: Divide both the sides of quadratic equation ax2 + bx + c = 0 by a. Now, the obtained equation is x2 + (b/a) x + c/a = 0
- Step 2: Subtract c/a from both the sides of quadratic equation x2 + (b/a) x + c/a = 0. Obtained equation is x2 + (b/a) x = -c/a
- Step 3: Add the square of (b/2a) to both the sides of quadratic equation x2 + (b/a) x = -c/a. Obtained equation is x2 + (b/a) x + (b/2a)2 = -c/a + (b/2a)2
- Step 4: Now the LHS of the quadratic equation x2 + (b/a) x + (b/2a)2 = -c/a + (b/2a)2 can be written as a complete square and simplify the RHS, if necessary. Obtained equation is (x + b/2a)2 = -c/a + (b/2a)2
- Step 5: The roots of the given quadratic equation can be obtained and hence, we can form the factors of the equation.
Another algebraic identity which is used for factoring quadratics is a2 - b2 = (a + b)(a - b). Let us see an example to understand.
Example 1: f(x) = 9x2 - 4 (difference of 2 perfect squares)
9x2 - 4 = (3x)2 - 22
We notice that this is of the form, a2 - b2 = (a + b)(a - b)
Hence we factorize the equation 9x2 - 4 = 0 as (3x+2) (3x-2)
9x2 - 4 = (3x+2) (3x-2)
Example 2: f(x) = 4x2 + 12x + 9
4x2 + 12x + 9 = (2x)2 + 2(2x)(3) + (3)2
We notice that this is of the form (a+b)2 = a2 + 2ab + b2
(2x)2 + 2(2x)(3) + (3)2 = (2x + 3)2
Hence we have (2x + 3), (2x + 3) as the linear factors of f(x) = 4x2 + 12x + 9
Formula for Factoring Quadratics
Factoring quadratics is also done by using a formula that gives us the roots of the quadratic equation and hence, the factors of the equation. If ax2 + bx + c = 0 is a quadratic equation, a is the coefficient of x2, b is the coefficient of x and c is the constant term. Then we find the value of x by using the formula:
\(x= \dfrac{-b\pm \sqrt{b^2-4ac}}{2a}\)
Consider, the quadratic equation x2 + 5x + 4 = 0. It is of the form ax2 + bx + c = 0. Here a =1, b = 5, c = 4
Substituting the values of a, b and c in the quadratic formula formula, we get
\[\begin{align}x&= \dfrac{-b\pm \sqrt{b^2-4ac}}{2a}\\\\x &= \dfrac{-5\pm \sqrt{5^2-4\times 1 \times 4}}{2\times 1}\\\\&= \dfrac{-5\pm \sqrt{25 -16}}{2}\\\\&= \dfrac{-5\pm \sqrt{9}}{2}\\\\&= \dfrac{-5\pm 3}{2}\\&= \dfrac{-5+3}{2}\text{ and } \dfrac{-5-3}{2}\\\\&=\dfrac{-2}{2}\text{ and }\dfrac{-8}{2}\\\\x &=(-1)\text{ and} (-4)\end{align}\]
Thus, the factors are (x + 1) and (x + 4).
Related Topics on Factoring Quadratics
- Factor Theorem
- Factors
- Factoring Methods
Important Notes on Factoring Quadratics
- Linear factors are of the form ax + b and they cannot be factored further.
- A quadratic polynomial is a polynomial of degree 2.
- The sum of the roots of the quadratic equation ax2 + bx + c = 0 is given by \(\alpha + \beta\) = -b/a
- The product of the roots in the quadratic equation ax2 + bx + c = 0 is given by \(\alpha\beta\) = c/a
Factoring Quadratics Examples
-
Example 1: Verify if (2x+3) and (x+3) are the linear factors of the quadratic equation f(x) = 2x2 + 9x + 9.
Solution: To verify if (2x+3) and (x+3) are the linear factors of the quadratic equation f(x) = 2x2 + 9x + 9, we will multiply the factors.
(2x + 3)(x + 3) = 2x2 + 3x + 6x + 9 = 2x2 + 9x + 9.
Answer: Hence, (2x+3) and (x+3) are the linear factors of the quadratic equation f(x) = 2x2 + 9x + 9.
-
Example 2: Find the factors of the quadratic equation x2 + x - 12 = 0 using the factoring quadratics method.
Solution: We will split the middle term of the quadratic equation x2 + x - 12 = 0 to determine its factors.
x2 + x - 12 = 0
⇒ x2 + 4x - 3x - 12 = 0
⇒ x(x + 4) - 3(x + 4) = 0
⇒ (x - 3)(x + 4) = 0
Answer: Hence the factors of x2 + x - 12 = 0 are (x - 3) and (x + 4).
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Practice Questions on Factoring Quadratics
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FAQs on Factoring Quadratics
What is Factoring Quadratics in Algebra?
Factoring quadratics is a method of expressing the polynomial as a product of its linear factors. It is a process that allows us to simplify quadratic expressions, find their roots and solve equations.
How to Solve Quadratic Equations by Factoring Quadratics?
Factoring quadratics gives us the roots of the quadratic equation. There are different methods that can be used for factoring quadratic equations. Factoring quadratics is done in 4 ways:
- Factoring out the GCD
- Splitting the middle term
- Using Algebraic Identities (Completing the Squares)
- Using Quadratic formula
By determining the factors, we can get the roots of the quadratic equation and hence the solution.
What are the Methods used for Factorization of Quadratic Equations?
The methods to factorize quadratic equations are splitting the middle term, using algebraic identities, using the quadratic formula, and factoring the GCD out.
What is the Method of Factoring Quadratic Equations?
Splitting the middle term and using the quadratic formula are the most efficient methods for factoring quadratic equations.
Is Factoring Quadratics the Same as Solving it?
When we factorize a quadratic equation, we get linear factors that divide the quadratic polynomial evenly. The next step is finding the zeros of the equation by equating the factors with zero.
What is the Shortest Way to Factor any Quadratic Equation?
Using the quadratic formula is the shortest way of factoring quadratics.
What Are Some Tricks for Factoring Quadratics?
Find the sum of the roots and the product of the roots or by identifying any known algebraic identity, we can factorize the quadratic equations.
What is Quadratic Equation Factored Form?
(x - \(\alpha\)) (x - \(\beta\)) is the factored form of quadratic equation, where \(\alpha\) and \(\beta\) are the roots of the quadratic equation.
How to Solve a Quadratic Equation?
There are different methods that can be used for factoring quadratic equations and solving the quadratic equations. Factoring quadratics is done in 4 ways:
- Factoring out the GCD
- Splitting the middle term
- Using Algebraic Identities (Completing the Squares)
- Using Quadratic formula
How Can you Factorize Quadratic Equations Easily?
We split the middle term b of the quadratic equation ax2 + bx + c = 0 when we try to factorize quadratic equations. We determine the factor pairs of the product of a and c such that their sum is equal to b. Taking the common factors out, we can factorize quadratic equations easily.
How to Compute the Bar Designation Using the Quadratic Equation
Source: https://www.cuemath.com/algebra/factorization-of-quadratic-equations/
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