Short Answer
To solve a quadratic equation using factorization, we first write the equation in standard form ax² + bx + c = 0. Then we break the expression into two factors such that their product equals the original equation.
After factorizing, we set each factor equal to zero and solve for the variable. This gives us the solutions (roots) of the quadratic equation.
Detailed Explanation
Solving quadratic equations using factorization
To solve a quadratic equation using the factorization method, we rewrite the equation into a product of two simple expressions. This makes it easy to find the values of the variable.
A quadratic equation is written as:
The goal is to change this equation into the form:
(x + p)(x + q) = 0
Once we get this form, we can easily find the solutions.
Steps to solve using factorization
Follow these simple steps:
- Write in standard form
Make sure the equation is written as ax² + bx + c = 0. - Find two numbers
Find two numbers whose sum is equal to b and product is equal to ac (or c when a = 1). - Split the middle term
Break the middle term (bx) using those two numbers. - Group the terms
Arrange the terms in pairs. - Factor each group
Take out common factors from each group. - Write in bracket form
Convert into (x + p)(x + q) = 0. - Apply zero product rule
If the product of two factors is zero, then at least one of them must be zero. - Solve each factor
Find the value of x by setting each bracket equal to zero.
Example for clear understanding
Let us solve the equation:
x² + 7x + 10 = 0
Step 1: Find two numbers whose sum is 7 and product is 10
These numbers are 5 and 2
Step 2: Factorize
(x + 5)(x + 2) = 0
Step 3: Solve
x + 5 = 0 → x = −5
x + 2 = 0 → x = −2
So, the solutions are x = −5 and x = −2.
Another example
Solve: 2x² + 5x + 3 = 0
Step 1: Multiply a and c → 2 × 3 = 6
Step 2: Find two numbers whose sum is 5 and product is 6 → 2 and 3
Step 3: Split middle term
2x² + 2x + 3x + 3 = 0
Step 4: Group terms
2x(x + 1) + 3(x + 1) = 0
Step 5: Factor
(2x + 3)(x + 1) = 0
Step 6: Solve
2x + 3 = 0 → x = −3/2
x + 1 = 0 → x = −1
Importance of factorization method
This method is useful because:
- It is simple and easy to apply
- It avoids long calculations
- It helps understand the structure of equations
- It is very useful for basic problems
However, not all quadratic equations can be easily factorized. In such cases, other methods like the quadratic formula are used.
Conclusion
Solving quadratic equations using factorization is a simple and effective method. It involves converting the equation into factors and then solving each factor separately. This method helps in quickly finding the solutions and builds a strong understanding of algebraic concepts.