Express { (x+10)^2$}$ As A Trinomial In Standard Form.

Introduction

In algebra, quadratic expressions are a fundamental concept that plays a crucial role in solving equations and inequalities. A quadratic expression is a polynomial of degree two, which means it has a highest power of two. In this article, we will focus on expressing the quadratic expression $(x+10)^2$ in standard form, also known as trinomial form.

What is a Trinomial?

A trinomial is a polynomial with three terms. In the context of quadratic expressions, a trinomial is a quadratic expression that can be written in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. The standard form of a quadratic expression is essential in solving quadratic equations and inequalities.

Expanding the Quadratic Expression

To express the quadratic expression $(x+10)^2$ in standard form, we need to expand it using the formula $(a+b)^2 = a^2 + 2ab + b^2$. In this case, $a = x$ and $b = 10$. Therefore, we can expand the expression as follows:

$$(x+10)^2 = x^2 + 2(x)(10) + 10^2$$

Simplifying the Expression

Now that we have expanded the expression, we can simplify it by evaluating the products and combining like terms. The expression becomes:

$$x^2 + 20x + 100$$

Standard Form of a Quadratic Expression

The standard form of a quadratic expression is $ax^2 + bx + c$. In this case, we have:

$$x^2 + 20x + 100$$

This is the standard form of the quadratic expression $(x+10)^2$. We can see that the coefficient of $x^2$ is 1, the coefficient of $x$ is 20, and the constant term is 100.

Why is Standard Form Important?

Standard form is essential in solving quadratic equations and inequalities. When we have a quadratic equation in standard form, we can use various techniques such as factoring, completing the square, and the quadratic formula to solve it. Additionally, standard form helps us to identify the vertex of a parabola, which is a critical concept in graphing and analyzing quadratic functions.

Conclusion

In this article, we have expressed the quadratic expression $(x+10)^2$ in standard form. We have used the formula $(a+b)^2 = a^2 + 2ab + b^2$ to expand the expression and then simplified it by evaluating the products and combining like terms. The standard form of a quadratic expression is essential in solving quadratic equations and inequalities, and it helps us to identify the vertex of a parabola.

Examples and Exercises

1. Express the quadratic expression $(x-5)^2$ in standard form.
2. Express the quadratic expression $(x+3)^2$ in standard form.
3. Solve the quadratic equation $x^2 + 6x + 8 = 0$ using the quadratic formula.
4. Find the vertex of the parabola $y = x^2 + 4x + 3$.

Answer Key

1. $x^2 - 10x + 25$
2. $x^2 + 6x + 9$
3. $x = -2$ or $x = -4$
4. The vertex is at the point $(-2, -1)$.

Glossary of Terms

• Quadratic expression: A polynomial of degree two.
• Trinomial: A polynomial with three terms.
• Standard form: A quadratic expression written in the form $ax^2 + bx + c$.
• Vertex: The point on a parabola where the function changes from increasing to decreasing or vice versa.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

About the Author

Introduction

In our previous article, we discussed how to express quadratic expressions in standard form. In this article, we will provide a Q&A guide to help you better understand quadratic expressions and how to work with them.

Q: What is a quadratic expression?

A: A quadratic expression is a polynomial of degree two, which means it has a highest power of two. It can be written in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Q: What is the standard form of a quadratic expression?

A: The standard form of a quadratic expression is $ax^2 + bx + c$. This is the form in which we typically write quadratic expressions.

Q: How do I expand a quadratic expression?

A: To expand a quadratic expression, you can use the formula $(a+b)^2 = a^2 + 2ab + b^2$. For example, to expand the expression $(x+10)^2$, you would use the formula as follows:

$$(x+10)^2 = x^2 + 2(x)(10) + 10^2$$

Q: How do I simplify a quadratic expression?

A: To simplify a quadratic expression, you can evaluate the products and combine like terms. For example, to simplify the expression $x^2 + 20x + 100$, you would combine the like terms as follows:

$$x^2 + 20x + 100 = x^2 + 20x + 100$$

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point on the parabola where the function changes from increasing to decreasing or vice versa. It is typically denoted by the point $(h, k)$, where $h$ is the x-coordinate and $k$ is the y-coordinate.

Q: How do I find the vertex of a parabola?

A: To find the vertex of a parabola, you can use the formula $h = -\frac{b}{2a}$ and $k = f(h)$. For example, to find the vertex of the parabola $y = x^2 + 4x + 3$, you would use the formula as follows:

$$h = -\frac{4}{2(1)} = -2$$

$$k = f(-2) = (-2)^2 + 4(-2) + 3 = -1$$

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to solve quadratic equations. It is given by:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Q: How do I use the quadratic formula to solve a quadratic equation?

A: To use the quadratic formula to solve a quadratic equation, you can plug in the values of $a$, $b$, and $c$ into the formula and simplify. For example, to solve the quadratic equation $x^2 + 6x + 8 = 0$, you would use the quadratic formula as follows:

$$x = \frac{-6 \pm \sqrt{6^2 - 4(1)(8)}}{2(1)}$$

$$x = \frac{-6 \pm \sqrt{36 - 32}}{2}$$

$$x = \frac{-6 \pm \sqrt{4}}{2}$$

$$x = \frac{-6 \pm 2}{2}$$

$$x = -4 \text{ or } x = -2$$

Conclusion

In this article, we have provided a Q&A guide to help you better understand quadratic expressions and how to work with them. We have covered topics such as expanding and simplifying quadratic expressions, finding the vertex of a parabola, and using the quadratic formula to solve quadratic equations.

Examples and Exercises

1. Expand the quadratic expression $(x-3)^2$.
2. Simplify the quadratic expression $x^2 + 12x + 36$.
3. Find the vertex of the parabola $y = x^2 + 6x + 5$.
4. Use the quadratic formula to solve the quadratic equation $x^2 + 8x + 12 = 0$.

Answer Key

1. $x^2 - 6x + 9$
2. $x^2 + 12x + 36$
3. The vertex is at the point $(-3, -2)$.
4. $x = -2$ or $x = -6$

Glossary of Terms

• Quadratic expression: A polynomial of degree two.
• Standard form: A quadratic expression written in the form $ax^2 + bx + c$.
• Vertex: The point on a parabola where the function changes from increasing to decreasing or vice versa.
• Quadratic formula: A formula that can be used to solve quadratic equations.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

About the Author

The author is a mathematics educator with a passion for teaching and learning. They have a strong background in algebra and calculus and have taught various mathematics courses at the high school and college levels.