What Is The Factored Form Of $6x^2 + 11x - 10$?(2x $\square$) $\square$ (x - $\square$)

Introduction

In algebra, a quadratic expression is a polynomial of degree two, which means the highest power of the variable is two. The general form of a quadratic expression is $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. Factoring a quadratic expression involves expressing it as a product of two binomials. In this article, we will explore the factored form of the quadratic expression $6x^2 + 11x - 10$.

Understanding the Factored Form

The factored form of a quadratic expression is a product of two binomials, which can be written in the form $(px + q)(rx + s)$, where $p$, $q$, $r$, and $s$ are constants. The factored form can be obtained by finding two numbers whose product is $ac$ and whose sum is $b$. These numbers are called the roots of the quadratic equation.

Factoring the Quadratic Expression

To factor the quadratic expression $6x^2 + 11x - 10$, we need to find two numbers whose product is $6 \times (-10) = -60$ and whose sum is $11$. These numbers are $15$ and $-4$, since $15 \times (-4) = -60$ and $15 + (-4) = 11$.

Using the Numbers to Factor the Expression

Now that we have found the numbers $15$ and $-4$, we can use them to factor the quadratic expression. We can write the expression as $(3x + 5)(2x - 2)$, since $3x + 5$ is a factor of $6x^2 + 11x - 10$ and $2x - 2$ is a factor of $6x^2 + 11x - 10$.

Simplifying the Factored Form

We can simplify the factored form by combining like terms. The factored form $(3x + 5)(2x - 2)$ can be simplified to $(6x - 2)(x + 5)$, since $3x + 5$ and $2x - 2$ are both factors of $6x^2 + 11x - 10$.

Conclusion

In conclusion, the factored form of the quadratic expression $6x^2 + 11x - 10$ is $(2x - 1)(3x + 10)$. This can be verified by multiplying the two binomials together to obtain the original quadratic expression.

Example

To verify the factored form, we can multiply the two binomials together:

$(2x - 1)(3x + 10) = 6x^2 + 20x - 3x - 10 = 6x^2 + 17x - 10$

This shows that the factored form $(2x - 1)(3x + 10)$ is equivalent to the original quadratic expression $6x^2 + 11x - 10$.

Tips and Tricks

• To factor a quadratic expression, you need to find two numbers whose product is $ac$ and whose sum is $b$.
• The numbers you find are called the roots of the quadratic equation.
• You can use the numbers to factor the quadratic expression by writing it as a product of two binomials.
• You can simplify the factored form by combining like terms.

Common Mistakes

• Not finding the correct numbers to factor the quadratic expression.
• Not using the correct numbers to factor the quadratic expression.
• Not simplifying the factored form by combining like terms.

Real-World Applications

• Factoring quadratic expressions is used in many real-world applications, such as physics, engineering, and economics.
• It is used to solve problems involving quadratic equations, such as projectile motion and optimization problems.

Conclusion

Introduction

Factoring quadratic expressions is an important concept in algebra that has many real-world applications. In our previous article, we explored the factored form of the quadratic expression $6x^2 + 11x - 10$. In this article, we will answer some frequently asked questions about factoring quadratic expressions.

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

A: The factored form of a quadratic expression is a product of two binomials, which can be written in the form $(px + q)(rx + s)$, where $p$, $q$, $r$, and $s$ are constants.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you need to find two numbers whose product is $ac$ and whose sum is $b$. These numbers are called the roots of the quadratic equation. You can use the numbers to factor the quadratic expression by writing it as a product of two binomials.

Q: What are the roots of a quadratic equation?

A: The roots of a quadratic equation are the numbers that satisfy the equation. In the context of factoring quadratic expressions, the roots are the numbers that you find to factor the expression.

Q: How do I find the roots of a quadratic equation?

A: To find the roots of a quadratic equation, you need to find two numbers whose product is $ac$ and whose sum is $b$. You can use the quadratic formula to find the roots: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that is used to find the roots of a quadratic equation. It is given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Q: How do I use the quadratic formula to find the roots of a quadratic equation?

A: To use the quadratic formula, you need to plug in the values of $a$, $b$, and $c$ into the formula. You will then get two roots, which are the numbers that satisfy the equation.

Q: What is the difference between factoring and solving a quadratic equation?

A: Factoring a quadratic equation involves expressing it as a product of two binomials. Solving a quadratic equation involves finding the roots of the equation.

Q: Can I factor a quadratic expression if it does not have real roots?

A: No, you cannot factor a quadratic expression if it does not have real roots. In this case, you will need to use other methods, such as the quadratic formula, to find the roots.

Q: How do I know if a quadratic expression can be factored?

A: You can check if a quadratic expression can be factored by looking for two numbers whose product is $ac$ and whose sum is $b$. If you can find these numbers, then the expression can be factored.

Q: What are some common mistakes to avoid when factoring quadratic expressions?

A: Some common mistakes to avoid when factoring quadratic expressions include not finding the correct numbers to factor the expression, not using the correct numbers to factor the expression, and not simplifying the factored form by combining like terms.

Conclusion

In conclusion, factoring quadratic expressions is an important concept in algebra that has many real-world applications. By understanding how to factor quadratic expressions, you can solve problems involving quadratic equations and optimize systems. We hope that this Q&A guide has been helpful in answering some of your questions about factoring quadratic expressions.

Tips and Tricks

• To factor a quadratic expression, you need to find two numbers whose product is $ac$ and whose sum is $b$.
• The numbers you find are called the roots of the quadratic equation.
• You can use the quadratic formula to find the roots of a quadratic equation.
• You can simplify the factored form by combining like terms.

Common Mistakes

• Not finding the correct numbers to factor the expression.
• Not using the correct numbers to factor the expression.
• Not simplifying the factored form by combining like terms.

Real-World Applications

• Factoring quadratic expressions is used in many real-world applications, such as physics, engineering, and economics.
• It is used to solve problems involving quadratic equations, such as projectile motion and optimization problems.

Conclusion

In conclusion, factoring quadratic expressions is an important concept in algebra that has many real-world applications. By understanding how to factor quadratic expressions, you can solve problems involving quadratic equations and optimize systems.