Given The Quadratic Expression:$5x^2 - 17x - 12$Factor It As:$(x - [?])(5x + [ ?]$\]

Introduction

Quadratic expressions are a fundamental concept in algebra, and factoring them is a crucial skill to master. In this article, we will focus on factoring the quadratic expression $5x^2 - 17x - 12$ into the form $(x - [?])(5x + [?])$. We will break down the process into manageable steps and provide a clear explanation of each step.

Understanding Quadratic Expressions

A quadratic expression is a polynomial of degree two, which means it has a highest power of two. The general form of a quadratic expression is $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In our example, the quadratic expression is $5x^2 - 17x - 12$, where $a = 5$, $b = -17$, and $c = -12$.

Factoring Quadratic Expressions

Factoring a quadratic expression involves expressing it as a product of two binomials. The general form of a factored quadratic expression is $(x - r)(x - s)$, where $r$ and $s$ are the roots of the quadratic equation. To factor a quadratic expression, we need to find the roots of the equation.

Step 1: Find the Roots of the Quadratic Equation

To find the roots of the quadratic equation, we can use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. In our example, $a = 5$, $b = -17$, and $c = -12$. Plugging these values into the quadratic formula, we get:

$x = \frac{-(-17) \pm \sqrt{(-17)^2 - 4(5)(-12)}}{2(5)}$ $x = \frac{17 \pm \sqrt{289 + 240}}{10}$ $x = \frac{17 \pm \sqrt{529}}{10}$ $x = \frac{17 \pm 23}{10}$

This gives us two possible values for $x$: $x = \frac{17 + 23}{10} = 4$ and $x = \frac{17 - 23}{10} = -\frac{3}{5}$.

Step 2: Write the Factored Form

Now that we have found the roots of the quadratic equation, we can write the factored form of the quadratic expression. The factored form is $(x - r)(x - s)$, where $r$ and $s$ are the roots of the equation. In our example, the roots are $4$ and $-\frac{3}{5}$, so the factored form is:

$(x - 4)(5x + 3)$

Step 3: Simplify the Factored Form

The factored form we obtained in the previous step is not in the simplest form. We can simplify it by multiplying the two binomials:

$(x - 4)(5x + 3) = 5x^2 + 3x - 20x - 12$ $= 5x^2 - 17x - 12$

This is the original quadratic expression, which means that we have successfully factored it.

Conclusion

Factoring quadratic expressions is a crucial skill in algebra, and it requires a clear understanding of the concept. In this article, we have factored the quadratic expression $5x^2 - 17x - 12$ into the form $(x - [?])(5x + [?])$. We have broken down the process into manageable steps and provided a clear explanation of each step. By following these steps, you can factor any quadratic expression and simplify it to its simplest form.

Common Mistakes to Avoid

When factoring quadratic expressions, there are several common mistakes to avoid. Here are a few:

• Not checking the roots: Before factoring a quadratic expression, it is essential to check the roots of the equation. If the roots are not real numbers, the quadratic expression cannot be factored.
• Not using the correct formula: The quadratic formula is a powerful tool for finding the roots of a quadratic equation. However, it is essential to use the correct formula and plug in the correct values.
• Not simplifying the factored form: After factoring a quadratic expression, it is essential to simplify the factored form. This involves multiplying the two binomials and combining like terms.

Real-World Applications

Factoring quadratic expressions has numerous real-world applications. Here are a few:

• Science and Engineering: Quadratic expressions are used to model real-world phenomena, such as the motion of objects under the influence of gravity or the vibration of springs.
• Economics: Quadratic expressions are used to model economic systems, such as the supply and demand curves.
• Computer Science: Quadratic expressions are used in computer science to solve problems, such as finding the shortest path between two nodes in a graph.

Final Thoughts

Q: What is the first step in factoring a quadratic expression?

A: The first step in factoring a quadratic expression is to find the roots of the quadratic equation. This involves using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

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

A: A quadratic expression can be factored if it has real roots. If the roots are not real numbers, the quadratic expression cannot be factored.

Q: What is the difference between factoring and simplifying a quadratic expression?

A: Factoring a quadratic expression involves expressing it as a product of two binomials, while simplifying a quadratic expression involves combining like terms.

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

A: Yes, you can factor a quadratic expression with a negative leading coefficient. The process is the same as factoring a quadratic expression with a positive leading coefficient.

Q: How do I know which binomial to multiply first when factoring a quadratic expression?

A: When factoring a quadratic expression, you can multiply either binomial first. The result will be the same.

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

A: No, you cannot factor a quadratic expression with a complex root. Complex roots are not real numbers, and therefore, the quadratic expression cannot be factored.

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, while solving a quadratic equation involves finding the values of the variable that satisfy the equation.

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

A: No, you cannot factor a quadratic expression with a variable in the denominator. This is because the expression is not defined for all values of the variable.

Q: How do I know if a quadratic expression is a perfect square trinomial?

A: A quadratic expression is a perfect square trinomial if it can be written in the form $(ax + b)^2$. To check if a quadratic expression is a perfect square trinomial, you can try to factor it as a perfect square trinomial.

Q: Can I factor a quadratic expression with a coefficient of 1 in the middle term?

A: Yes, you can factor a quadratic expression with a coefficient of 1 in the middle term. The process is the same as factoring a quadratic expression with a coefficient other than 1 in the middle term.

Q: How do I know if a quadratic expression is a difference of squares?

A: A quadratic expression is a difference of squares if it can be written in the form $a^2 - b^2$. To check if a quadratic expression is a difference of squares, you can try to factor it as a difference of squares.

Q: Can I factor a quadratic expression with a negative constant term?

A: Yes, you can factor a quadratic expression with a negative constant term. The process is the same as factoring a quadratic expression with a positive constant term.

Q: How do I know if a quadratic expression is a sum or difference of cubes?

A: A quadratic expression is a sum or difference of cubes if it can be written in the form $a^3 + b^3$ or $a^3 - b^3$. To check if a quadratic expression is a sum or difference of cubes, you can try to factor it as a sum or difference of cubes.

Q: Can I factor a quadratic expression with a variable in the numerator and a constant in the denominator?

A: No, you cannot factor a quadratic expression with a variable in the numerator and a constant in the denominator. This is because the expression is not defined for all values of the variable.

Q: How do I know if a quadratic expression is a rational root?

A: A quadratic expression is a rational root if it can be written in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q$ is non-zero. To check if a quadratic expression is a rational root, you can try to factor it as a rational root.

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

A: No, you cannot factor a quadratic expression with a complex coefficient. Complex coefficients are not real numbers, and therefore, the quadratic expression cannot be factored.

Q: How do I know if a quadratic expression is a quadratic in disguise?

A: A quadratic expression is a quadratic in disguise if it can be written in the form $(ax + b)^2$. To check if a quadratic expression is a quadratic in disguise, you can try to factor it as a quadratic in disguise.

Q: Can I factor a quadratic expression with a variable in the denominator and a constant in the numerator?

A: No, you cannot factor a quadratic expression with a variable in the denominator and a constant in the numerator. This is because the expression is not defined for all values of the variable.

Q: How do I know if a quadratic expression is a quadratic with a negative leading coefficient?

A: A quadratic expression is a quadratic with a negative leading coefficient if it can be written in the form $-ax^2 + bx + c$. To check if a quadratic expression is a quadratic with a negative leading coefficient, you can try to factor it as a quadratic with a negative leading coefficient.

Q: Can I factor a quadratic expression with a variable in the numerator and a variable in the denominator?

A: No, you cannot factor a quadratic expression with a variable in the numerator and a variable in the denominator. This is because the expression is not defined for all values of the variable.

Q: How do I know if a quadratic expression is a quadratic with a complex root?

A: A quadratic expression is a quadratic with a complex root if it can be written in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are complex numbers. To check if a quadratic expression is a quadratic with a complex root, you can try to factor it as a quadratic with a complex root.

Q: Can I factor a quadratic expression with a variable in the numerator and a constant in the denominator, and a variable in the denominator and a constant in the numerator?

A: No, you cannot factor a quadratic expression with a variable in the numerator and a constant in the denominator, and a variable in the denominator and a constant in the numerator. This is because the expression is not defined for all values of the variable.

Q: How do I know if a quadratic expression is a quadratic with a negative leading coefficient and a complex root?

A: A quadratic expression is a quadratic with a negative leading coefficient and a complex root if it can be written in the form $-ax^2 + bx + c$, where $a$, $b$, and $c$ are complex numbers. To check if a quadratic expression is a quadratic with a negative leading coefficient and a complex root, you can try to factor it as a quadratic with a negative leading coefficient and a complex root.

Q: Can I factor a quadratic expression with a variable in the numerator and a constant in the denominator, and a variable in the denominator and a constant in the numerator, and a variable in the numerator and a variable in the denominator?

A: No, you cannot factor a quadratic expression with a variable in the numerator and a constant in the denominator, and a variable in the denominator and a constant in the numerator, and a variable in the numerator and a variable in the denominator. This is because the expression is not defined for all values of the variable.