Not simplifying the expression inside the square root

Oct 13, 2025 by miodeya.com 16 views

Introduction

The quadratic formula is a powerful tool for solving quadratic equations of the form $ax^2 + bx + c = 0$ . It is a general method that can be used to find the solutions to any quadratic equation, regardless of whether the equation can be factored or not. In this article, we will use the quadratic formula to solve the equation $x^2 + 2x - 2 = 0$ .

What is the Quadratic Formula?

The quadratic formula is given by:

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

where $a$ , $b$ , and $c$ are the coefficients of the quadratic equation. In the equation $x^2 + 2x - 2 = 0$ , we have $a = 1$ , $b = 2$ , and $c = -2$ .

Step 1: Plug in the Values

To use the quadratic formula, we need to plug in the values of $a$ , $b$ , and $c$ into the formula. In this case, we have:

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

Step 2: Simplify the Expression

Now, we need to simplify the expression inside the square root. We have:

2^2 - 4(1)(-2) = 4 + 8 = 12

So, the expression becomes:

x = \frac{-2 \pm \sqrt{12}}{2}

Step 3: Simplify the Square Root

The square root of 12 can be simplified as:

\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}

So, the expression becomes:

x = \frac{-2 \pm 2\sqrt{3}}{2}

Step 4: Simplify the Expression

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

x = \frac{-2 \pm 2\sqrt{3}}{2} = -1 \pm \sqrt{3}

Conclusion

We have used the quadratic formula to solve the equation $x^2 + 2x - 2 = 0$ . The solutions to the equation are $x = -1 + \sqrt{3}$ and $x = -1 - \sqrt{3}$ .

Why is the Quadratic Formula Important?

The quadratic formula is an important tool in mathematics because it can be used to solve quadratic equations that cannot be factored. It is a general method that can be used to find the solutions to any quadratic equation, regardless of whether the equation can be factored or not.

Real-World Applications of the Quadratic Formula

The quadratic formula has many real-world applications. For example, it can be used to model the trajectory of a projectile, to find the maximum height of a thrown object, and to determine the time it takes for an object to reach its maximum height.

Common Mistakes to Avoid

When using the quadratic formula, there are several common mistakes to avoid. These include:

Not simplifying the expression inside the square root: Make sure to simplify the expression inside the square root before plugging it into the formula.
Not simplifying the square root: Make sure to simplify the square root before plugging it into the formula.
Not simplifying the expression: Make sure to simplify the expression after plugging in the values.

Conclusion

In conclusion, the quadratic formula is a powerful tool for solving quadratic equations. It is a general method that can be used to find the solutions to any quadratic equation, regardless of whether the equation can be factored or not. By following the steps outlined in this article, you can use the quadratic formula to solve quadratic equations and simplify your answers.

Additional Resources

For more information on the quadratic formula, check out the following resources:

Mathway: A online math problem solver that can be used to solve quadratic equations.
Khan Academy: A online learning platform that offers video lessons on quadratic equations.
Wolfram Alpha: A online calculator that can be used to solve quadratic equations.

Final Answer

$The final answer is: $\boxed{-1 + \sqrt{3}, -1 - \sqrt{3}}$<br/> **Quadratic Formula Q&A**$

Frequently Asked Questions

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations of the form $ax^2 + bx + c = 0$ . It is a general method that can be used to find the solutions to any quadratic equation, regardless of whether the equation can be factored or not.

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of $a$ , $b$ , and $c$ into the formula. The formula is given by:

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

Q: What is the difference between the quadratic formula and factoring?

A: The quadratic formula and factoring are two different methods for solving quadratic equations. Factoring involves finding two binomials whose product is equal to the original equation, while the quadratic formula involves using a formula to find the solutions to the equation.

Q: Can I use the quadratic formula to solve any quadratic equation?

A: Yes, you can use the quadratic formula to solve any quadratic equation, regardless of whether the equation can be factored or not.

Q: What is the significance of the discriminant in the quadratic formula?

A: The discriminant is the expression inside the square root in the quadratic formula, which is given by $b^2 - 4ac$ . The discriminant determines the nature of the solutions to the equation. 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.

Q: Can I use the quadratic formula to solve quadratic equations with complex solutions?

A: Yes, you can use the quadratic formula to solve quadratic equations with complex solutions. The quadratic formula will give you the complex solutions to the equation.

Q: How do I simplify the expression inside the square root in the quadratic formula?

A: To simplify the expression inside the square root, you need to factor the expression and simplify it. For example, if the expression inside the square root is $12$ , you can factor it as $4 \cdot 3$ and simplify it to $2\sqrt{3}$ .

Q: Can I use the quadratic formula to solve quadratic equations with rational coefficients?

A: Yes, you can use the quadratic formula to solve quadratic equations with rational coefficients. The quadratic formula will give you the rational solutions to the equation.

Q: What are some common mistakes to avoid when using the quadratic formula?

A: Some common mistakes to avoid when using the quadratic formula include:

Not simplifying the expression inside the square root: Make sure to simplify the expression inside the square root before plugging it into the formula.
Not simplifying the square root: Make sure to simplify the square root before plugging it into the formula.
Not simplifying the expression: Make sure to simplify the expression after plugging in the values.

Q: Can I use the quadratic formula to solve quadratic equations with negative coefficients?

A: Yes, you can use the quadratic formula to solve quadratic equations with negative coefficients. The quadratic formula will give you the solutions to the equation.

Q: How do I check my solutions to a quadratic equation?

A: To check your solutions to a quadratic equation, you need to plug the solutions back into the original equation and simplify it. If the solutions satisfy the equation, then they are the correct solutions.

Conclusion

Additional Resources

For more information on the quadratic formula, check out the following resources:

Mathway: A online math problem solver that can be used to solve quadratic equations.
Khan Academy: A online learning platform that offers video lessons on quadratic equations.
Wolfram Alpha: A online calculator that can be used to solve quadratic equations.

Final Answer

The final answer is: $\boxed{Yes, you can use the quadratic formula to solve any quadratic equation, regardless of whether the equation can be factored or not.}$