Factoring And Solving Quadratic Equations Worksheet

Understanding Quadratic Equations

Quadratic equations are polynomial equations of degree two, which means the highest power of the variable is two. These equations can be written in the general form of ax² + bx + c = 0, where a, b, and c are constants, and x is the variable. Solving quadratic equations is a crucial skill in algebra and is used to solve problems in various fields, including physics, engineering, and computer science.

Factoring Quadratic Equations

One way to solve quadratic equations is by factoring. Factoring involves expressing the quadratic equation as a product of two binomials. The factored form of a quadratic equation can be written as (x + m)(x + n) = 0, where m and n are constants.

To factor a quadratic equation, we need to find two numbers whose product is c and whose sum is b. These numbers are then used to write the factored form of the equation.

📝 Note: Not all quadratic equations can be factored. In such cases, other methods, such as the quadratic formula, need to be used.

Steps to Factor Quadratic Equations

Here are the steps to factor quadratic equations:

1. Write the quadratic equation in the standard form ax² + bx + c = 0.
2. Look for two numbers whose product is c and whose sum is b. These numbers are called the factors of c and b.
3. Write the factored form of the equation using the factors found in step 2.

Examples of Factoring Quadratic Equations

Example 1: Factor the quadratic equation x² + 5x + 6 = 0.

Solution: We need to find two numbers whose product is 6 and whose sum is 5. The numbers are 2 and 3. Therefore, the factored form of the equation is (x + 2)(x + 3) = 0.

Example 2: Factor the quadratic equation x² - 7x + 12 = 0.

Solution: We need to find two numbers whose product is 12 and whose sum is -7. The numbers are -3 and -4. Therefore, the factored form of the equation is (x - 3)(x - 4) = 0.

Solving Quadratic Equations

Once the quadratic equation is factored, it can be solved by setting each factor equal to zero and solving for the variable.

Example 1: Solve the quadratic equation (x + 2)(x + 3) = 0.

Solution: Set each factor equal to zero and solve for x:

x + 2 = 0 –> x = -2

x + 3 = 0 –> x = -3

Therefore, the solutions to the equation are x = -2 and x = -3.

Example 2: Solve the quadratic equation (x - 3)(x - 4) = 0.

Solution: Set each factor equal to zero and solve for x:

x - 3 = 0 –> x = 3

x - 4 = 0 –> x = 4

Therefore, the solutions to the equation are x = 3 and x = 4.

📝 Note: The solutions to a quadratic equation can be real or complex numbers.

Quadratic Formula

If a quadratic equation cannot be factored, it can be solved using the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

Where a, b, and c are the coefficients of the quadratic equation.

Conclusion

Factoring and solving quadratic equations are essential skills in algebra. By following the steps outlined in this article, you can factor and solve quadratic equations with ease. Remember to use the quadratic formula if the equation cannot be factored.