Solve The Following Equation:${ Y^2 = 10y - 16 }$

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Introduction

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Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving the equation $y^2 = 10y - 16$. This equation is a quadratic equation in the form of $ax^2 + bx + c = 0$, where $a = 1$, $b = -10$, and $c = 16$. We will use various methods to solve this equation, including factoring, the quadratic formula, and completing the square.

Understanding Quadratic Equations

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A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. Quadratic equations can be solved using various methods, including factoring, the quadratic formula, and completing the square.

Factoring Quadratic Equations

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Factoring is a method of solving quadratic equations by expressing them as a product of two binomials. To factor a quadratic equation, we need to find two numbers whose product is equal to the constant term ($c$) and whose sum is equal to the coefficient of the linear term ($b$). Once we have these two numbers, we can write the quadratic equation as a product of two binomials.

The Quadratic Formula

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The quadratic formula is a method of solving quadratic equations that is based on the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. This formula can be used to solve any quadratic equation, regardless of whether it can be factored or not.

Completing the Square

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Completing the square is a method of solving quadratic equations by rewriting them in the form $(x + \frac{b}{2a})^2 = \frac{c}{a}$. This method is useful for solving quadratic equations that cannot be factored.

Solving the Equation $y^2 = 10y - 16$

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Now that we have discussed the various methods of solving quadratic equations, let's apply these methods to the equation $y^2 = 10y - 16$. We can start by rearranging the equation to get $y^2 - 10y + 16 = 0$.

Factoring the Equation

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We can try to factor the equation $y^2 - 10y + 16 = 0$ by finding two numbers whose product is equal to the constant term ($16$) and whose sum is equal to the coefficient of the linear term ($-10$). These two numbers are $-8$ and $-2$, since $(-8) \times (-2) = 16$ and $(-8) + (-2) = -10$. Therefore, we can write the equation as $(y - 8)(y - 2) = 0$.

Solving for $y$

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To solve for $y$, we can set each factor equal to zero and solve for $y$. This gives us two possible solutions: $y - 8 = 0$ and $y - 2 = 0$. Solving for $y$ in each of these equations gives us $y = 8$ and $y = 2$.

Using the Quadratic Formula

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We can also use the quadratic formula to solve the equation $y^2 - 10y + 16 = 0$. The quadratic formula is $y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. In this case, $a = 1$, $b = -10$, and $c = 16$. Plugging these values into the formula gives us $y = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(1)(16)}}{2(1)}$. Simplifying this expression gives us $y = \frac{10 \pm \sqrt{100 - 64}}{2}$. This simplifies further to $y = \frac{10 \pm \sqrt{36}}{2}$, which gives us $y = \frac{10 \pm 6}{2}$. Therefore, we have two possible solutions: $y = \frac{10 + 6}{2} = 8$ and $y = \frac{10 - 6}{2} = 2$.

Completing the Square

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We can also use completing the square to solve the equation $y^2 - 10y + 16 = 0$. To do this, we need to rewrite the equation in the form $(y + \frac{b}{2a})^2 = \frac{c}{a}$. In this case, $a = 1$, $b = -10$, and $c = 16$. Plugging these values into the formula gives us $(y + \frac{-10}{2(1)})^2 = \frac{16}{1}$. Simplifying this expression gives us $(y - 5)^2 = 16$. Taking the square root of both sides gives us $y - 5 = \pm 4$. Therefore, we have two possible solutions: $y = 5 + 4 = 9$ and $y = 5 - 4 = 1$.

Conclusion

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In this article, we have discussed the various methods of solving quadratic equations, including factoring, the quadratic formula, and completing the square. We have applied these methods to the equation $y^2 = 10y - 16$ and found two possible solutions: $y = 8$ and $y = 2$. These solutions can be verified by plugging them back into the original equation.

Final Answer

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The final answer is $\boxed{8}$ and $\boxed{2}$.

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Introduction

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Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will answer some of the most frequently asked questions about quadratic equations, including factoring, the quadratic formula, and completing the square.

Q: What is a quadratic equation?

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A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: How do I factor a quadratic equation?

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A: To factor a quadratic equation, you need to find two numbers whose product is equal to the constant term ($c$) and whose sum is equal to the coefficient of the linear term ($b$). Once you have these two numbers, you can write the quadratic equation as a product of two binomials.

Q: What is the quadratic formula?

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A: The quadratic formula is a method of solving quadratic equations that is based on the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. This formula can be used to solve any quadratic equation, regardless of whether it can be factored or not.

Q: How do I use the quadratic formula?

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A: To use the quadratic formula, you need to plug in the values of $a$, $b$, and $c$ into the formula. Then, you need to simplify the expression and solve for $x$.

Q: What is completing the square?

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A: Completing the square is a method of solving quadratic equations by rewriting them in the form $(x + \frac{b}{2a})^2 = \frac{c}{a}$. This method is useful for solving quadratic equations that cannot be factored.

Q: How do I complete the square?

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A: To complete the square, you need to rewrite the quadratic equation in the form $(x + \frac{b}{2a})^2 = \frac{c}{a}$. Then, you need to take the square root of both sides and solve for $x$.

Q: What are the steps to solve a quadratic equation?

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A: The steps to solve a quadratic equation are:

1. Check if the equation can be factored: If the equation can be factored, use factoring to solve for $x$.
2. Use the quadratic formula: If the equation cannot be factored, use the quadratic formula to solve for $x$.
3. Complete the square: If the equation cannot be factored and the quadratic formula is not applicable, use completing the square to solve for $x$.

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

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

• Not checking if the equation can be factored: Make sure to check if the equation can be factored before using the quadratic formula or completing the square.
• Not using the correct formula: Make sure to use the correct formula for the method you are using.
• Not simplifying the expression: Make sure to simplify the expression before solving for $x$.
• Not checking for extraneous solutions: Make sure to check for extraneous solutions when using the quadratic formula or completing the square.

Conclusion

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In this article, we have answered some of the most frequently asked questions about quadratic equations, including factoring, the quadratic formula, and completing the square. We hope that this article has been helpful in clarifying some of the concepts and methods involved in solving quadratic equations.

Final Answer

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The final answer is that quadratic equations are a fundamental concept in mathematics, and solving them requires a combination of factoring, the quadratic formula, and completing the square. By following the steps outlined in this article, you can solve quadratic equations with confidence.