Which Is The Completely Factored Form Of $r^2 + 2r + 10$?

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Introduction

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In algebra, factoring quadratic expressions is a crucial skill that helps us simplify complex equations and solve problems more efficiently. A quadratic expression is a polynomial of degree two, which means it has a highest power of two. Factoring a quadratic expression involves expressing it as a product of two binomials or other quadratic expressions. In this article, we will explore the completely factored form of the quadratic expression $r^2 + 2r + 10$.

Understanding the Quadratic Expression

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The given quadratic expression is $r^2 + 2r + 10$. To factor this expression, we need to find two numbers whose product is $10$ and whose sum is $2$. These numbers are $2$ and $5$, because $2 \times 5 = 10$ and $2 + 5 = 7$, but we need a sum of 2, so we will use a different method.

Factoring the Quadratic Expression

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To factor the quadratic expression $r^2 + 2r + 10$, we can use the method of completing the square. This method involves rewriting the quadratic expression in a form that allows us to easily factor it.

Completing the Square

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To complete the square, we need to add and subtract a constant term to the quadratic expression. The constant term is the square of half the coefficient of the linear term. In this case, the coefficient of the linear term is $2$, so half of it is $1$, and the square of $1$ is $1$. Therefore, we add and subtract $1$ to the quadratic expression:

$r^2 + 2r + 10 = (r^2 + 2r + 1) + 10 - 1$

Simplifying the Expression

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Now, we can simplify the expression by combining like terms:

$(r^2 + 2r + 1) + 10 - 1 = (r + 1)^2 + 9$

Factoring the Expression

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Finally, we can factor the expression by recognizing that it is a sum of two squares:

$(r + 1)^2 + 9 = (r + 1)^2 + 3^2$

Conclusion

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In conclusion, the completely factored form of the quadratic expression $r^2 + 2r + 10$ is $(r + 1)^2 + 3^2$. This form is obtained by completing the square and recognizing that it is a sum of two squares.

Example Problems

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Problem 1

Factor the quadratic expression $x^2 + 4x + 4$.

Solution

To factor the quadratic expression $x^2 + 4x + 4$, we can use the method of completing the square. We add and subtract $4$ to the quadratic expression:

$x^2 + 4x + 4 = (x^2 + 4x + 4) - 4 + 4$

We can then simplify the expression by combining like terms:

$(x^2 + 4x + 4) - 4 + 4 = (x + 2)^2$

Therefore, the completely factored form of the quadratic expression $x^2 + 4x + 4$ is $(x + 2)^2$.

Problem 2

Factor the quadratic expression $y^2 - 6y + 9$.

Solution

To factor the quadratic expression $y^2 - 6y + 9$, we can use the method of completing the square. We add and subtract $9$ to the quadratic expression:

$y^2 - 6y + 9 = (y^2 - 6y + 9) - 9 + 9$

We can then simplify the expression by combining like terms:

$(y^2 - 6y + 9) - 9 + 9 = (y - 3)^2$

Therefore, the completely factored form of the quadratic expression $y^2 - 6y + 9$ is $(y - 3)^2$.

Tips and Tricks

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Tip 1

When factoring a quadratic expression, always look for two numbers whose product is the constant term and whose sum is the coefficient of the linear term.

Tip 2

If the quadratic expression cannot be factored using the method of factoring, try completing the square.

Tip 3

When completing the square, always add and subtract the same constant term to the quadratic expression.

Conclusion

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In conclusion, factoring quadratic expressions is a crucial skill in algebra that helps us simplify complex equations and solve problems more efficiently. By using the method of completing the square, we can factor quadratic expressions that cannot be factored using the method of factoring. Remember to always look for two numbers whose product is the constant term and whose sum is the coefficient of the linear term, and to add and subtract the same constant term when completing the square. With practice and patience, you will become proficient in factoring quadratic expressions and solving problems more efficiently.

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Introduction

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In our previous article, we explored the completely factored form of the quadratic expression $r^2 + 2r + 10$. We also discussed the method of completing the square and how it can be used to factor quadratic expressions. In this article, we will answer some frequently asked questions about factoring quadratic expressions.

Q&A

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Q: What is the difference between factoring and completing the square?

A: Factoring involves expressing a quadratic expression as a product of two binomials or other quadratic expressions, while completing the square involves rewriting a quadratic expression in a form that allows us to easily factor it.

Q: How do I know when to use factoring and when to use completing the square?

A: If the quadratic expression can be easily factored using the method of factoring, then use factoring. If the quadratic expression cannot be factored using the method of factoring, then try completing the square.

Q: What is the formula for completing the square?

A: The formula for completing the square is:

$(x + \frac{b}{2})^2 - (\frac{b}{2})^2 + c$

where $x$ is the variable, $b$ is the coefficient of the linear term, and $c$ is the constant term.

Q: How do I factor a quadratic expression with a negative leading coefficient?

A: To factor a quadratic expression with a negative leading coefficient, first factor out the negative sign, then factor the remaining quadratic expression.

Q: Can I factor a quadratic expression with a complex coefficient?

A: Yes, you can factor a quadratic expression with a complex coefficient. However, the factored form may involve complex numbers.

Q: How do I factor a quadratic expression with a variable in the denominator?

A: To factor a quadratic expression with a variable in the denominator, first simplify the expression by multiplying the numerator and denominator by the conjugate of the denominator.

Q: Can I factor a quadratic expression with a fractional coefficient?

A: Yes, you can factor a quadratic expression with a fractional coefficient. However, the factored form may involve fractions.

Example Problems

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Problem 1

Factor the quadratic expression $x^2 + 6x + 8$.

Solution

To factor the quadratic expression $x^2 + 6x + 8$, we can use the method of factoring. We look for two numbers whose product is $8$ and whose sum is $6$. These numbers are $4$ and $2$, because $4 \times 2 = 8$ and $4 + 2 = 6$. Therefore, the completely factored form of the quadratic expression $x^2 + 6x + 8$ is $(x + 4)(x + 2)$.

Problem 2

Factor the quadratic expression $y^2 - 4y + 4$.

Solution

To factor the quadratic expression $y^2 - 4y + 4$, we can use the method of completing the square. We add and subtract $4$ to the quadratic expression:

$y^2 - 4y + 4 = (y^2 - 4y + 4) - 4 + 4$

We can then simplify the expression by combining like terms:

$(y^2 - 4y + 4) - 4 + 4 = (y - 2)^2$

Therefore, the completely factored form of the quadratic expression $y^2 - 4y + 4$ is $(y - 2)^2$.

Tips and Tricks

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Tip 1

When factoring a quadratic expression, always look for two numbers whose product is the constant term and whose sum is the coefficient of the linear term.

Tip 2

If the quadratic expression cannot be factored using the method of factoring, try completing the square.

Tip 3

When completing the square, always add and subtract the same constant term to the quadratic expression.

Conclusion

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In conclusion, factoring quadratic expressions is a crucial skill in algebra that helps us simplify complex equations and solve problems more efficiently. By using the method of factoring and completing the square, we can factor quadratic expressions that cannot be factored using the method of factoring. Remember to always look for two numbers whose product is the constant term and whose sum is the coefficient of the linear term, and to add and subtract the same constant term when completing the square. With practice and patience, you will become proficient in factoring quadratic expressions and solving problems more efficiently.