Simplify The Expression:${ 7y^2 - 20y - 3 }$
Introduction
In mathematics, simplifying expressions is a crucial step in solving equations and inequalities. It involves rewriting an expression in a more compact and manageable form, often by combining like terms or factoring out common factors. In this article, we will simplify the given expression: 7y^2 - 20y - 3.
Understanding the Expression
Before we simplify the expression, let's break it down and understand its components. The expression consists of three terms:
- 7y^2: This is a quadratic term, where 7 is the coefficient and y^2 is the variable.
- -20y: This is a linear term, where -20 is the coefficient and y is the variable.
- -3: This is a constant term.
Simplifying the Expression
To simplify the expression, we can start by factoring out the greatest common factor (GCF) of the three terms. In this case, the GCF is 1, since there is no common factor that can be factored out.
However, we can try to factor the quadratic term 7y^2. We can look for two numbers whose product is 7 and whose sum is -20. These numbers are -7 and 3, since (-7)(3) = -21 and -7 + 3 = -4. However, this does not match our expression.
Another approach is to use the method of grouping. We can group the first two terms together and factor out the common factor:
7y^2 - 20y = 7y(y - 3)
Now, we can add the constant term -3 to the expression:
7y(y - 3) - 3
However, this does not simplify the expression further.
Using the Quadratic Formula
Since we were unable to factor the quadratic term 7y^2, we can use the quadratic formula to simplify the expression. The quadratic formula is:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 7, b = -20, and c = -3. Plugging these values into the formula, we get:
y = (20 ± √((-20)^2 - 4(7)(-3))) / 2(7) y = (20 ± √(400 + 84)) / 14 y = (20 ± √484) / 14 y = (20 ± 22) / 14
Simplifying the expression, we get two possible values for y:
y = (20 + 22) / 14 = 42 / 14 = 3 y = (20 - 22) / 14 = -2 / 14 = -1/7
Conclusion
In this article, we simplified the expression 7y^2 - 20y - 3 using various methods, including factoring and the quadratic formula. We found that the expression cannot be factored further, but we were able to simplify it using the quadratic formula. The simplified expression is:
y = 3 or y = -1/7
Final Answer
Introduction
In our previous article, we simplified the expression 7y^2 - 20y - 3 using various methods, including factoring and the quadratic formula. In this article, we will answer some frequently asked questions (FAQs) related to the simplification of the expression.
Q&A
Q: What is the greatest common factor (GCF) of the three terms in the expression 7y^2 - 20y - 3?
A: The GCF of the three terms is 1, since there is no common factor that can be factored out.
Q: Can we factor the quadratic term 7y^2?
A: Yes, we can try to factor the quadratic term 7y^2. However, we need to find two numbers whose product is 7 and whose sum is -20. These numbers are -7 and 3, but this does not match our expression.
Q: What is the method of grouping?
A: The method of grouping is a technique used to factor quadratic expressions. We can group the first two terms together and factor out the common factor.
Q: Can we use the quadratic formula to simplify the expression 7y^2 - 20y - 3?
A: Yes, we can use the quadratic formula to simplify the expression. The quadratic formula is:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 7, b = -20, and c = -3. Plugging these values into the formula, we get:
y = (20 ± √((-20)^2 - 4(7)(-3))) / 2(7) y = (20 ± √(400 + 84)) / 14 y = (20 ± √484) / 14 y = (20 ± 22) / 14
Simplifying the expression, we get two possible values for y:
y = (20 + 22) / 14 = 42 / 14 = 3 y = (20 - 22) / 14 = -2 / 14 = -1/7
Q: What is the final answer to the expression 7y^2 - 20y - 3?
A: The final answer to the expression 7y^2 - 20y - 3 is:
y = 3 or y = -1/7
Q: Can we factor the expression 7y^2 - 20y - 3 further?
A: No, we cannot factor the expression 7y^2 - 20y - 3 further. The expression has been simplified using the quadratic formula.
Q: What is the significance of the quadratic formula in simplifying expressions?
A: The quadratic formula is a powerful tool used to simplify quadratic expressions. It can be used to find the solutions to quadratic equations and to simplify expressions.
Conclusion
In this article, we answered some frequently asked questions (FAQs) related to the simplification of the expression 7y^2 - 20y - 3. We discussed the greatest common factor (GCF), factoring, the method of grouping, the quadratic formula, and the final answer to the expression. We hope that this article has been helpful in understanding the simplification of the expression.
Final Answer
The final answer is: