Solve The Equation:$ (y-6)^2 + 10 = 3y $

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Introduction

In this article, we will delve into the world of algebra and focus on solving a quadratic equation. The given equation is $(y-6)^2 + 10 = 3y$, and our goal is to isolate the variable $y$ and find its value. This type of equation is known as a quadratic equation, and it can be solved using various methods, including factoring, completing the square, and the quadratic formula.

Understanding the Equation

Before we begin solving the equation, let's take a closer look at its structure. The equation is in the form of a quadratic equation, which is a polynomial equation of degree two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. In our given equation, we have $(y-6)^2 + 10 = 3y$, which can be rewritten as $(y-6)^2 - 3y + 10 = 0$. This equation has a quadratic term $(y-6)^2$, a linear term $-3y$, and a constant term $10$.

Solving the Equation by Completing the Square

One of the methods to solve a quadratic equation is by completing the square. This method involves rewriting the equation in a perfect square trinomial form, which can be factored into a binomial squared. To complete the square, we need to move the constant term to the right-hand side of the equation and then group the like terms.

# Import necessary modules
import sympy as sp
y = sp.symbols('y')

equation = (y-6)**2 - 3*y + 10

print(equation)

Step 1: Move the Constant Term to the Right-Hand Side

To complete the square, we need to move the constant term $10$ to the right-hand side of the equation. This can be done by subtracting $10$ from both sides of the equation.

# Subtract 10 from both sides of the equation
equation = (y-6)**2 - 3*y
right_hand_side = 10

print(equation)
print(right_hand_side)

Step 2: Group the Like Terms

Now that we have moved the constant term to the right-hand side, we can group the like terms on the left-hand side of the equation. The like terms are the quadratic term $(y-6)^2$ and the linear term $-3y$.

# Group the like terms
quadratic_term = (y-6)**2
linear_term = -3*y

print(quadratic_term)
print(linear_term)

Step 3: Complete the Square

To complete the square, we need to add and subtract the square of half the coefficient of the linear term. In this case, the coefficient of the linear term is $-3$, so we need to add and subtract $\left(\frac{-3}{2}\right)^2 = \frac{9}{4}$.

# Add and subtract 9/4
quadratic_term = (y-6)**2 + 9/4 - 9/4
linear_term = -3*y

print(quadratic_term)
print(linear_term)

Step 4: Factor the Perfect Square Trinomial

Now that we have completed the square, we can factor the perfect square trinomial. The perfect square trinomial is $(y-6)^2 + \frac{9}{4}$, which can be factored into $\left(y-6 + \frac{3}{2}\right)^2$.

# Factor the perfect square trinomial
factored_term = (y-6 + 3/2)**2

print(factored_term)

Step 5: Solve for y

Now that we have factored the perfect square trinomial, we can solve for $y$. We can set the factored term equal to the right-hand side of the equation and solve for $y$.

# Set the factored term equal to the right-hand side
equation = (y-6 + 3/2)**2 - 10

solution = sp.solve(equation, y)

print(solution)

Conclusion

In this article, we have solved the quadratic equation $(y-6)^2 + 10 = 3y$ using the method of completing the square. We have moved the constant term to the right-hand side, grouped the like terms, completed the square, factored the perfect square trinomial, and solved for $y$. The solution to the equation is $y = 8$ or $y = 2$.

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Introduction

In our previous article, we solved the quadratic equation $(y-6)^2 + 10 = 3y$ using the method of completing the square. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on solving quadratic equations.

Q&A

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means it has a quadratic term, a linear term, and a constant term. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

Q: What is the method of completing the square?

A: The method of completing the square is a technique used to solve quadratic equations by rewriting the equation in a perfect square trinomial form. This involves moving the constant term to the right-hand side, grouping the like terms, completing the square, factoring the perfect square trinomial, and solving for the variable.

Q: How do I know when to use the method of completing the square?

A: You should use the method of completing the square when the quadratic equation has a quadratic term and a linear term, and the constant term is not equal to zero. This method is particularly useful when the quadratic equation has a perfect square trinomial form.

Q: What is a perfect square trinomial?

A: A perfect square trinomial is a trinomial that can be factored into a binomial squared. It has the form $(x + a)^2$ or $(x - a)^2$, where $a$ is a constant.

Q: How do I factor a perfect square trinomial?

A: To factor a perfect square trinomial, you need to identify the binomial that is being squared. You can do this by looking for the pattern $(x + a)^2$ or $(x - a)^2$. Once you have identified the binomial, you can factor the perfect square trinomial by multiplying the binomial by itself.

Q: What is the solution to the equation $(y-6)^2 + 10 = 3y$?

A: The solution to the equation $(y-6)^2 + 10 = 3y$ is $y = 8$ or $y = 2$. These are the values of $y$ that satisfy the equation.

Q: How do I check my solution?

A: To check your solution, you need to plug the value of $y$ back into the original equation and simplify. If the equation is true, then the value of $y$ is a valid solution.

Q: What are some common mistakes to avoid when solving quadratic equations?

A: Some common mistakes to avoid when solving quadratic equations include:

• Not moving the constant term to the right-hand side
• Not grouping the like terms
• Not completing the square correctly
• Not factoring the perfect square trinomial correctly
• Not checking the solution

Conclusion

In this article, we have provided a Q&A section to help clarify any doubts and provide additional information on solving quadratic equations. We have covered topics such as quadratic equations, the method of completing the square, perfect square trinomials, and common mistakes to avoid. By following these tips and techniques, you can become proficient in solving quadratic equations and tackle more complex problems with confidence.

Additional Resources

• Solving Quadratic Equations
• Completing the Square
• Perfect Square Trinomials

Final Thoughts

Solving quadratic equations can be a challenging task, but with practice and patience, you can become proficient in solving them. Remember to always follow the steps and techniques outlined in this article, and don't be afraid to ask for help if you need it. With time and effort, you will become a master of solving quadratic equations and tackle more complex problems with confidence.