Bailey Writes The Expression G 2 + 14 G + 40 G^2 + 14g + 40 G 2 + 14 G + 40 To Represent The Area Of A Planned School Garden In Square Feet. What Factors Can Be Used To Find The Dimensions Of Her Garden?A. ( G − 4 ) ( G − 10 (g - 4)(g - 10 ( G − 4 ) ( G − 10 ] B. ( G + 4 ) ( G + 10 (g + 4)(g + 10 ( G + 4 ) ( G + 10 ] C.
Understanding Quadratic Expressions
In mathematics, a quadratic expression is a polynomial of degree two, which means the highest power of the variable is two. These expressions are commonly represented in the form of ax^2 + bx + c, where a, b, and c are constants. In the given problem, Bailey has expressed the area of her school garden as a quadratic expression: g^2 + 14g + 40.
Factoring Quadratic Expressions
To find the dimensions of Bailey's garden, we need to factor the quadratic expression g^2 + 14g + 40. Factoring a quadratic expression involves expressing it as a product of two binomials. The general form of factoring a quadratic expression is (x - p)(x - q), where p and q are the roots of the equation.
Finding the Factors
To find the factors of the quadratic expression g^2 + 14g + 40, we need to look for two numbers whose product is 40 and whose sum is 14. These numbers are 10 and 4, as 10 * 4 = 40 and 10 + 4 = 14.
Writing the Factored Form
Using the numbers 10 and 4, we can write the factored form of the quadratic expression as (g - 10)(g - 4). This is because the product of the two binomials (g - 10) and (g - 4) gives us the original quadratic expression g^2 + 14g + 40.
Choosing the Correct Factor
Now that we have factored the quadratic expression, we need to choose the correct factor. The correct factor is the one that represents the dimensions of the garden. In this case, the correct factor is (g - 10)(g - 4).
Conclusion
In conclusion, to find the dimensions of Bailey's garden, we need to factor the quadratic expression g^2 + 14g + 40. By finding the factors (g - 10)(g - 4), we can determine the length and width of the garden. The length of the garden is 10 feet, and the width is 4 feet.
Answer
The correct answer is A. (g - 4)(g - 10).
Additional Information
- The quadratic expression g^2 + 14g + 40 can also be factored as (g + 4)(g + 10), but this is not the correct answer.
- The factored form of the quadratic expression is (g - 10)(g - 4), which represents the dimensions of the garden.
- The length of the garden is 10 feet, and the width is 4 feet.
Final Answer
Q: What is a quadratic expression?
A: A quadratic expression is a polynomial of degree two, which means the highest power of the variable is two. These expressions are commonly represented in the form of ax^2 + bx + c, where a, b, and c are constants.
Q: How do I factor a quadratic expression?
A: To factor a quadratic expression, you need to express it as a product of two binomials. The general form of factoring a quadratic expression is (x - p)(x - q), where p and q are the roots of the equation.
Q: What are the roots of a quadratic equation?
A: The roots of a quadratic equation are the values of x that make the equation true. In other words, they are the solutions to the equation.
Q: How do I find the roots of a quadratic equation?
A: To find the roots of a quadratic equation, you need to factor the equation or use the quadratic formula. The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a.
Q: What is the quadratic formula?
A: The quadratic formula is a formula that gives the solutions to a quadratic equation. It is x = (-b ± √(b^2 - 4ac)) / 2a.
Q: How do I use the quadratic formula?
A: To use the quadratic formula, you need to plug in the values of a, b, and c into the formula. Then, simplify the expression and solve for x.
Q: What is the difference between factoring and using the quadratic formula?
A: Factoring involves expressing a quadratic expression as a product of two binomials, while using the quadratic formula involves plugging in values into a formula to find the solutions to a quadratic equation.
Q: When should I use factoring and when should I use the quadratic formula?
A: You should use factoring when the quadratic expression can be easily factored, and you should use the quadratic formula when the quadratic expression cannot be easily factored.
Q: Can I use both factoring and the quadratic formula to solve a quadratic equation?
A: Yes, you can use both factoring and the quadratic formula to solve a quadratic equation. However, factoring is usually the preferred method because it is often easier and faster.
Q: What are some common mistakes to avoid when factoring quadratic expressions?
A: Some common mistakes to avoid when factoring quadratic expressions include:
- Not checking if the expression can be factored
- Not using the correct method for factoring (e.g. using the quadratic formula when factoring is possible)
- Not simplifying the expression after factoring
- Not checking if the factored form is correct
Q: How do I check if a factored form is correct?
A: To check if a factored form is correct, you need to multiply the two binomials together and simplify the expression. If the result is the original quadratic expression, then the factored form is correct.
Q: What are some real-world applications of factoring quadratic expressions?
A: Some real-world applications of factoring quadratic expressions include:
- Finding the dimensions of a garden or a room
- Determining the maximum or minimum value of a function
- Solving problems involving motion or projectile motion
- Modeling population growth or decline
Q: Can I use factoring to solve systems of equations?
A: Yes, you can use factoring to solve systems of equations. However, this is usually more complicated and may require the use of other methods, such as substitution or elimination.
Q: What are some tips for factoring quadratic expressions?
A: Some tips for factoring quadratic expressions include:
- Look for common factors
- Use the quadratic formula as a last resort
- Check if the expression can be factored by grouping
- Use technology, such as a calculator or computer program, to help with factoring
Q: Can I use factoring to solve polynomial equations of degree three or higher?
A: No, you cannot use factoring to solve polynomial equations of degree three or higher. These equations require more advanced methods, such as the rational root theorem or synthetic division.