Which Shows The Correct Substitution Of The Values \[$a, B\$\], And \[$c\$\] From The Equation \[$0=4x^2+2x-1\$\] Into The Quadratic Formula Below?Quadratic Formula: \[$x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}\$\]A.

Understanding the Quadratic Formula

The quadratic formula is a powerful tool used to solve quadratic equations of the form ${ax^2+bx+c=0}$. It is a general formula that can be applied to any quadratic equation, regardless of the values of ${a}$, ${b}$, and ${c}$. The quadratic formula is given by:

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

Identifying the Values of ${a}$, ${b}$, and ${c}$

In the given equation ${0=4x^2+2x-1}$, we can identify the values of ${a}$, ${b}$, and ${c}$ as follows:

• ${a=4}$
• ${b=2}$
• ${c=-1}$

Substituting the Values into the Quadratic Formula

To solve the equation ${0=4x^2+2x-1}$, we need to substitute the values of ${a}$, ${b}$, and ${c}$ into the quadratic formula. This will give us:

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

Simplifying the Expression

Now, let's simplify the expression inside the square root:

${2^2-4(4)(-1)=4+16=20\$}

So, the expression becomes:

{x=\frac{-2 \pm \sqrt{20}}{8}$}$

Further Simplification

We can further simplify the expression by factoring out a perfect square from the square root:

{\sqrt{20}=\sqrt{4 \cdot 5}=2\sqrt{5}$}$

So, the expression becomes:

{x=\frac{-2 \pm 2\sqrt{5}}{8}$}$

Final Simplification

Finally, we can simplify the expression by dividing both the numerator and the denominator by 2:

{x=\frac{-1 \pm \sqrt{5}}{4}$}$

Conclusion

In conclusion, the correct substitution of the values ${a=4}$, ${b=2}$, and ${c=-1}$ from the equation ${0=4x^2+2x-1}$ into the quadratic formula is:

{x=\frac{-1 \pm \sqrt{5}}{4}$}$

This is the solution to the quadratic equation ${0=4x^2+2x-1}$.

Common Mistakes to Avoid

When substituting values into the quadratic formula, it's essential to be careful with the signs and the order of operations. Here are some common mistakes to avoid:

• Incorrect signs: Make sure to use the correct signs for the values of ${a}$, ${b}$, and ${c}$.
• Incorrect order of operations: Follow the order of operations (PEMDAS) when simplifying the expression inside the square root.
• Failure to simplify: Don't forget to simplify the expression by factoring out perfect squares and dividing by common factors.

Real-World Applications

The quadratic formula has numerous real-world applications in various fields, 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, and to optimize resource allocation.

Conclusion

In conclusion, the quadratic formula is a powerful tool for solving quadratic equations. By substituting the values of ${a}$, ${b}$, and ${c}$ into the formula, we can find the solutions to quadratic equations. Remember to be careful with signs and order of operations, and don't forget to simplify the expression by factoring out perfect squares and dividing by common factors. The quadratic formula has numerous real-world applications in various fields, including physics, engineering, and economics.

What is the Quadratic Formula?

The quadratic formula is a mathematical formula used to solve quadratic equations of the form ${ax^2+bx+c=0}$. It is a general formula that can be applied to any quadratic equation, regardless of the values of ${a}$, ${b}$, and ${c}$.

What is the Quadratic Formula Equation?

The quadratic formula equation is given by:

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

How Do I Use the Quadratic Formula?

To use the quadratic formula, you need to substitute the values of ${a}$, ${b}$, and ${c}$ into the formula. This will give you the solutions to the quadratic equation.

What are the Values of ${a}$, ${b}$, and ${c}$ in the Quadratic Formula?

In the quadratic formula, ${a}$ is the coefficient of the squared term, ${b}$ is the coefficient of the linear term, and ${c}$ is the constant term.

How Do I Simplify the Expression Inside the Square Root?

To simplify the expression inside the square root, you need to factor out a perfect square from the expression. This will give you a simplified expression that is easier to work with.

What is the Difference Between the Quadratic Formula and Factoring?

The quadratic formula and factoring are two different methods for solving quadratic equations. The quadratic formula is a general formula that can be applied to any quadratic equation, while factoring is a method that involves finding the factors of the quadratic expression.

When Should I Use the Quadratic Formula?

You should use the quadratic formula when:

• The quadratic expression cannot be factored: If the quadratic expression cannot be factored, the quadratic formula is the best method to use.
• The quadratic equation has complex solutions: If the quadratic equation has complex solutions, the quadratic formula is the best method to use.
• The quadratic equation has multiple solutions: If the quadratic equation has multiple solutions, the quadratic formula is the best method to use.

What are the Advantages of the Quadratic Formula?

The advantages of the quadratic formula include:

• It can be applied to any quadratic equation: The quadratic formula can be applied to any quadratic equation, regardless of the values of ${a}$, ${b}$, and ${c}$.
• It is a general formula: The quadratic formula is a general formula that can be applied to any quadratic equation.
• It is easy to use: The quadratic formula is easy to use and requires minimal calculations.

What are the Disadvantages of the Quadratic Formula?

The disadvantages of the quadratic formula include:

• It can be complex: The quadratic formula can be complex and difficult to understand.
• It requires calculations: The quadratic formula requires calculations, which can be time-consuming and error-prone.
• It may not be the best method for all quadratic equations: The quadratic formula may not be the best method for all quadratic equations, especially those that can be factored easily.

Conclusion

In conclusion, the quadratic formula is a powerful tool for solving quadratic equations. It is a general formula that can be applied to any quadratic equation, regardless of the values of ${a}$, ${b}$, and ${c}$. While it has its advantages and disadvantages, the quadratic formula is an essential tool for anyone who needs to solve quadratic equations.

Frequently Asked Questions

Here are some frequently asked questions about the quadratic formula:

• Q: What is the quadratic formula? A: The quadratic formula is a mathematical formula used to solve quadratic equations of the form ${ax^2+bx+c=0}$.
• Q: How do I use the quadratic formula? A: To use the quadratic formula, you need to substitute the values of ${a}$, ${b}$, and ${c}$ into the formula.
• Q: What are the values of ${a}$, ${b}$, and ${c}$ in the quadratic formula? A: In the quadratic formula, ${a}$ is the coefficient of the squared term, ${b}$ is the coefficient of the linear term, and ${c}$ is the constant term.
• Q: How do I simplify the expression inside the square root? A: To simplify the expression inside the square root, you need to factor out a perfect square from the expression.

Additional Resources

Here are some additional resources that you may find helpful:

• Quadratic Formula Calculator: A quadratic formula calculator can help you solve quadratic equations quickly and easily.
• Quadratic Formula Tutorial: A quadratic formula tutorial can provide you with a step-by-step guide on how to use the quadratic formula.
• Quadratic Formula Examples: Quadratic formula examples can help you understand how to use the quadratic formula in different situations.

Conclusion

In conclusion, the quadratic formula is a powerful tool for solving quadratic equations. It is a general formula that can be applied to any quadratic equation, regardless of the values of ${a}$, ${b}$, and ${c}$. While it has its advantages and disadvantages, the quadratic formula is an essential tool for anyone who needs to solve quadratic equations.