Answer Key For Solving Quadratic Equations By Factoring
Table of Contents :
Solving quadratic equations can often seem daunting at first, but using the method of factoring simplifies the process significantly. In this article, we'll delve into the steps for solving these equations, provide an answer key, and clarify some important concepts along the way. Let’s get started! 📚
What are Quadratic Equations?
A quadratic equation is a polynomial equation of the form:
[ ax^2 + bx + c = 0 ]
Where:
- (a), (b), and (c) are constants,
- (x) is the variable we want to solve for.
Quadratic equations can be solved through various methods including factoring, completing the square, or using the quadratic formula. In this article, we'll focus on the factoring method.
The Factoring Method
Factoring involves rewriting the quadratic equation as a product of two binomials. The equation will look like this:
[ (ax + m)(bx + n) = 0 ]
To find the roots of the quadratic equation using factoring, follow these steps:
- Rearrange the equation: Ensure that your equation is set to equal zero.
- Factor the quadratic: Rewrite the quadratic in the form of two binomials.
- Set each factor equal to zero: Solve for (x) from each factor.
- Check your solutions: Substitute the solutions back into the original equation to verify.
Example of Factoring a Quadratic Equation
Let’s consider a couple of examples for clarity.
Example 1:
Solve the quadratic equation:
[ x^2 + 5x + 6 = 0 ]
Step 1: Factor the quadratic
To factor (x^2 + 5x + 6), we need two numbers that multiply to 6 (the constant term) and add up to 5 (the coefficient of (x)).
These numbers are 2 and 3.
So, we can factor the equation as:
[ (x + 2)(x + 3) = 0 ]
Step 2: Set each factor to zero
Now, we set each factor equal to zero:
- (x + 2 = 0 \Rightarrow x = -2)
- (x + 3 = 0 \Rightarrow x = -3)
Solutions: (x = -2) and (x = -3) ✅
Example 2:
Solve the quadratic equation:
[ 2x^2 + 8x = 0 ]
Step 1: Rearrange the equation
First, we move everything to one side of the equation:
[ 2x^2 + 8x = 0 ]
Step 2: Factor out common terms
We can factor out a 2x:
[ 2x(x + 4) = 0 ]
Step 3: Set each factor to zero
Now we set each factor to zero:
- (2x = 0 \Rightarrow x = 0)
- (x + 4 = 0 \Rightarrow x = -4)
Solutions: (x = 0) and (x = -4) ✅
Answer Key for Common Quadratic Equations
Here’s a summary table of solutions to various quadratic equations solved by factoring:
<table> <tr> <th>Quadratic Equation</th> <th>Factored Form</th> <th>Solutions</th> </tr> <tr> <td>x² + 5x + 6 = 0</td> <td>(x + 2)(x + 3) = 0</td> <td>x = -2, x = -3</td> </tr> <tr> <td>2x² + 8x = 0</td> <td>2x(x + 4) = 0</td> <td>x = 0, x = -4</td> </tr> <tr> <td>x² - 9 = 0</td> <td>(x - 3)(x + 3) = 0</td> <td>x = 3, x = -3</td> </tr> <tr> <td>x² + 7x + 10 = 0</td> <td>(x + 2)(x + 5) = 0</td> <td>x = -2, x = -5</td> </tr> <tr> <td>x² - 5x + 6 = 0</td> <td>(x - 2)(x - 3) = 0</td> <td>x = 2, x = 3</td> </tr> </table>
Tips and Important Notes
- Identify Common Factors: Always look for common factors that can be factored out first.
- Use the Zero Product Property: Once factored, use the zero product property, which states that if the product of two factors is zero, at least one of the factors must be zero.
- Double-check your work: Substitute your solutions back into the original equation to ensure accuracy.
Important Note: "Factoring works best when the quadratic is set to zero and the equation can be easily factored into binomials. If it cannot be factored easily, other methods such as the quadratic formula may be necessary."
Conclusion
Factoring quadratic equations may seem challenging, but with practice and familiarity with common forms, it can become an efficient method for finding roots. Remember the steps outlined, refer to the answer key, and keep practicing with different quadratic equations. With time, you will find yourself solving these equations with confidence! 🌟