A) Solve The Equation: ${ 3x^2 + 45x + 150 = 0 }$

Introduction

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 quadratic equation $3x^2 + 45x + 150 = 0$. We will break down the solution into manageable steps, using algebraic techniques and formulas to find the roots of the equation.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, $x$) is two. The general form of a quadratic equation is:

$$ax^2 + bx + c = 0$$

where $a$, $b$, and $c$ are constants, and $a \neq 0$. In our equation, $a = 3$, $b = 45$, and $c = 150$.

Factoring the Quadratic Equation

One way to solve a quadratic equation is to factor it, if possible. Factoring involves expressing the quadratic expression as a product of two binomials. In this case, we can try to factor the equation by finding two numbers whose product is $3 \times 150 = 450$ and whose sum is $45$. These numbers are $25$ and $18$, since $25 \times 18 = 450$ and $25 + 18 = 43$. However, we need to find two numbers that add up to 45, so we can try to factor the equation as:

$$3x^2 + 45x + 150 = (3x + 25)(x + 6) = 0$$

Using the Zero Product Property

The zero product property states that if the product of two or more factors is zero, then at least one of the factors must be zero. In this case, we have:

$$(3x + 25)(x + 6) = 0$$

This means that either $3x + 25 = 0$ or $x + 6 = 0$.

Solving for x

Let's solve for $x$ in each of the two equations:

Solving $3x + 25 = 0$

Subtracting $25$ from both sides gives:

$$3x = -25$$

Dividing both sides by $3$ gives:

$$x = -\frac{25}{3}$$

Solving $x + 6 = 0$

Subtracting $6$ from both sides gives:

$$x = -6$$

Conclusion

We have solved the quadratic equation $3x^2 + 45x + 150 = 0$ using factoring and the zero product property. The solutions are $x = -\frac{25}{3}$ and $x = -6$. These are the values of $x$ that make the equation true.

Real-World Applications

Quadratic equations have many real-world applications, including:

• Physics: Quadratic equations are used to model the motion of objects under the influence of gravity, friction, and other forces.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
• Economics: Quadratic equations are used to model economic systems, including supply and demand curves.

Tips and Tricks

Here are some tips and tricks for solving quadratic equations:

• Use factoring: Factoring is a powerful technique for solving quadratic equations. Try to factor the equation before using other methods.
• Use the zero product property: The zero product property is a useful tool for solving quadratic equations. Use it to find the values of $x$ that make the equation true.
• Check your work: Always check your work by plugging the solutions back into the original equation.

Common Mistakes

Here are some common mistakes to avoid when solving quadratic equations:

• Not factoring: Failing to factor the equation can make it difficult to solve.
• Not using the zero product property: Failing to use the zero product property can lead to incorrect solutions.
• Not checking work: Failing to check your work can lead to incorrect solutions.

Conclusion

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Introduction

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.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, $x$) is two. The general form of a quadratic equation is:

$$ax^2 + bx + c = 0$$

where $a$, $b$, and $c$ are constants, and $a \neq 0$.

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including:

• Factoring: Factoring involves expressing the quadratic expression as a product of two binomials.
• Using the quadratic formula: The quadratic formula is a formula that can be used to find the solutions of a quadratic equation.
• Graphing: Graphing involves plotting the quadratic function on a graph and finding the x-intercepts.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to find the solutions of a quadratic equation. The formula is:

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

where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

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

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. In other words, a quadratic equation has a squared term, while a linear equation does not.

Q: Can I use a calculator to solve a quadratic equation?

A: Yes, you can use a calculator to solve a quadratic equation. Most calculators have a built-in quadratic formula function that can be used to find the solutions of a quadratic equation.

Q: What are some real-world applications of quadratic equations?

A: Quadratic equations have many real-world applications, including:

• Physics: Quadratic equations are used to model the motion of objects under the influence of gravity, friction, and other forces.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
• Economics: Quadratic equations are used to model economic systems, including supply and demand curves.

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

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

• Not factoring: Failing to factor the equation can make it difficult to solve.
• Not using the quadratic formula: Failing to use the quadratic formula can lead to incorrect solutions.
• Not checking work: Failing to check your work can lead to incorrect solutions.

Q: How do I know if a quadratic equation has real solutions?

A: To determine if a quadratic equation has real solutions, you can use the discriminant, which is the expression under the square root in the quadratic formula. If the discriminant is positive, the equation has two real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.

Conclusion

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we have answered some of the most frequently asked questions about quadratic equations. We hope this article has been helpful in clarifying any doubts you may have had about quadratic equations.