How To Factorization Quadratic Equation
Factoring a quadratic equation involves expressing it in the form of (ax^2 + bx + c) = 0, where a, b, and c are constants and x is the variable. The goal is to find two factors that, when multiplied together, give the quadratic equation in this form.
Here is the general process for factoring a quadratic equation and how to factorization quadratic equation:
1. Write the quadratic equation in the form of ax^2 + bx + c = 0.
2. Determine the values of a, b, and c.
3. Determine whether the equation can be factored using the difference of squares or the sum/difference of cubes method.
4. If the equation can be factored using the difference of squares method, factor it by finding two numbers whose product is equal to c and whose sum is equal to b.
5. If the equation can be factored using the sum/difference of cubes method, factor it by finding two numbers whose product is equal to b and whose sum/difference is equal to a.
6. If the equation cannot be factored using either of these methods, it may be necessary to use the quadratic formula to solve it.
For example, consider the quadratic equation x^2 - 5x + 6 = 0. To factor this equation, we can use the difference of squares method. The product of the two factors is 6 (the value of c), and their sum is -5 (the value of b). We can find two numbers that meet these criteria by using trial and error, or by using a list of factor pairs. In this case, the two numbers are 3 and 2, so the equation can be factored as (x - 3)(x - 2) = 0.
Here are some examples of factoring quadratic equations, along with a table showing the values of a, b, and c for each equation:
| Equation | a | b | c |
|---|
| x^2 + 2x + 1 = 0 | 1 | 2 | 1 |
| x^2 - 3x - 4 = 0 | 1 | -3 | -4 |
| x^2 + x - 6 = 0 | 1 | 1 | -6 |
To factor the first equation, x^2 + 2x + 1 = 0, we can use the difference of squares method. The product of the two factors is 1 (the value of c), and their sum is 2 (the value of b). We can find two numbers that meet these criteria by using trial and error, or by using a list of factor pairs. In this case, the two numbers are 1 and 1, so the equation can be factored as (x + 1)(x + 1) = 0.
To factor the second equation, x^2 - 3x - 4 = 0, we can use the difference of squares method. The product of the two factors is -4 (the value of c), and their sum is -3 (the value of b). We can find two numbers that meet these criteria by using trial and error, or by using a list of factor pairs. In this case, the two numbers are -2 and 2, so the equation can be factored as (x - 2)(x + 2) = 0.
To factor the third equation, x^2 + x - 6 = 0, we can use the sum/difference of cubes method. The product of the two factors is -6 (the value of c), and their sum/difference is 1 (the value of a). We can find two numbers that meet these criteria by using trial and error, or by using a list of factor pairs. In this case, the two numbers are -3 and -2, so the equation can be factored as (x - 3)(x - 2) = 0.
Factorization Quadratic Equation Examples
here are some examples of factoring quadratic equations, along with the steps for each example:
Example 1: Factor the equation x^2 + 2x + 1 = 0
Write the quadratic equation in the form of ax^2 + bx + c = 0. In this case, a = 1, b = 2, and c = 1.
Determine whether the equation can be factored using the difference of squares or the sum/difference of cubes method. In this case, it can be factored using the difference of squares method.
Find two numbers whose product is equal to c and whose sum is equal to b. In this case, the product of the two numbers is 1 (the value of c), and their sum is 2 (the value of b). We can find two numbers that meet these criteria by using trial and error, or by using a list of factor pairs.
Factor the equation using the two numbers found in step 3. In this case, the two numbers are 1 and 1, so the equation can be factored as (x + 1)(x + 1) = 0.
Example 2: Factor the equation x^2 - 3x - 4 = 0
Write the quadratic equation in the form of ax^2 + bx + c = 0. In this case, a = 1, b = -3, and c = -4.
Determine whether the equation can be factored using the difference of squares or the sum/difference of cubes method. In this case, it can be factored using the difference of squares method.
Find two numbers whose product is equal to c and whose sum is equal to b. In this case, the product of the two numbers is -4 (the value of c), and their sum is -3 (the value of b). We can find two numbers that meet these criteria by using trial and error, or by using a list of factor pairs.
Factor the equation using the two numbers found in step 3. In this case, the two numbers are -2 and 2, so the equation can be factored as (x - 2)(x + 2) = 0.
Example 3: Factor the equation x^2 + x - 6 = 0
Write the quadratic equation in the form of ax^2 + bx + c = 0. In this case, a = 1, b = 1, and c = -6.
Determine whether the equation can be factored using the difference of squares or the sum/difference of cubes method. In this case, it can be factored using the sum/difference of cubes method.
Find two numbers whose product is equal to c and whose sum/difference is equal to a. In this case, the product of the two numbers is -6 (the value of c), and their sum/difference is 1 (the value of a). We can find two numbers that meet these criteria by using trial and error, or by using a list of factor pairs.
Factor the equation using the two numbers found in step 3. In this case, the two numbers are -3 and -2, so the equation can be factored as (x - 3)(x - 2) = 0.
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