Solve $2x^2 + X - 4 = 0$.To Complete The Square:1. Start With $x^2 + \frac{1}{2}x$.2. Add And Subtract The Square Of Half The Coefficient Of $x$ (i.e., $\left(\frac{1}{4}\right)^2 = \frac{1}{16}$): $x^2 +

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. One of the methods used to solve quadratic equations is completing the square. In this article, we will explore the process of completing the square and apply it to solve the quadratic equation $2x^2 + x - 4 = 0$.

What is Completing the Square?

Completing the square is a technique used to solve quadratic equations by rewriting them in a perfect square form. This method involves manipulating the equation to create a perfect square trinomial, which can be factored into the square of a binomial. The process of completing the square is based on the concept of adding and subtracting a constant term to create a perfect square.

Step 1: Start with the Given Quadratic Equation

The given quadratic equation is $2x^2 + x - 4 = 0$. To complete the square, we need to start by isolating the $x^2$ term. We can do this by dividing both sides of the equation by 2, which gives us:

$$x^2 + \frac{1}{2}x - 2 = 0$$

Step 2: Add and Subtract the Square of Half the Coefficient of $x$

The next step is to add and subtract the square of half the coefficient of $x$. In this case, the coefficient of $x$ is $\frac{1}{2}$, and half of this coefficient is $\frac{1}{4}$. The square of $\frac{1}{4}$ is $\left(\frac{1}{4}\right)^2 = \frac{1}{16}$. We add and subtract $\frac{1}{16}$ to the equation:

$$x^2 + \frac{1}{2}x + \frac{1}{16} - \frac{1}{16} - 2 = 0$$

Step 3: Create a Perfect Square Trinomial

Now, we can create a perfect square trinomial by combining the first three terms of the equation:

$$\left(x + \frac{1}{4}\right)^2 - \frac{1}{16} - 2 = 0$$

Step 4: Simplify the Equation

We can simplify the equation by combining the constant terms:

$$\left(x + \frac{1}{4}\right)^2 - \frac{33}{16} = 0$$

Step 5: Add $\frac{33}{16}$ to Both Sides

To isolate the perfect square trinomial, we need to add $\frac{33}{16}$ to both sides of the equation:

$$\left(x + \frac{1}{4}\right)^2 = \frac{33}{16}$$

Step 6: Take the Square Root of Both Sides

Finally, we can take the square root of both sides of the equation to solve for $x$:

$$x + \frac{1}{4} = \pm\sqrt{\frac{33}{16}}$$

Simplifying the Solution

We can simplify the solution by evaluating the square root:

$$x + \frac{1}{4} = \pm\frac{\sqrt{33}}{4}$$

Subtracting $\frac{1}{4}$ from Both Sides

To isolate $x$, we need to subtract $\frac{1}{4}$ from both sides of the equation:

$$x = -\frac{1}{4} \pm \frac{\sqrt{33}}{4}$$

Conclusion

In this article, we have demonstrated the process of completing the square to solve the quadratic equation $2x^2 + x - 4 = 0$. By following the steps outlined above, we have successfully rewritten the equation in a perfect square form and solved for $x$. This method can be applied to solve a wide range of quadratic equations, and it is an essential tool for students and professionals in mathematics and related fields.

Additional Resources

For further practice and review, we recommend the following resources:

• Khan Academy: Quadratic Equations
• Mathway: Quadratic Equation Solver
• Wolfram Alpha: Quadratic Equation Solver

Final Thoughts

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Q: What is completing the square?

A: Completing the square is a technique used to solve quadratic equations by rewriting them in a perfect square form. This method involves manipulating the equation to create a perfect square trinomial, which can be factored into the square of a binomial.

Q: Why is completing the square useful?

A: Completing the square is a useful technique for solving quadratic equations because it allows us to rewrite the equation in a form that is easier to work with. This method can be used to solve a wide range of quadratic equations, and it is an essential tool for students and professionals in mathematics and related fields.

Q: How do I know when to use completing the square?

A: You should use completing the square when you are given a quadratic equation in the form of $ax^2 + bx + c = 0$, and you want to solve for $x$. This method is particularly useful when the equation is not easily factorable.

Q: What are the steps involved in completing the square?

A: The steps involved in completing the square are:

1. Start with the given quadratic equation.
2. Add and subtract the square of half the coefficient of $x$.
3. Create a perfect square trinomial.
4. Simplify the equation.
5. Take the square root of both sides to solve for $x$.

Q: What is the difference between completing the square and factoring?

A: Completing the square and factoring are two different methods for solving quadratic equations. Factoring involves expressing the quadratic equation as a product of two binomials, while completing the square involves rewriting the equation in a perfect square form.

Q: Can I use completing the square to solve all types of quadratic equations?

A: No, completing the square is not suitable for all types of quadratic equations. This method is particularly useful for quadratic equations that are not easily factorable, but it may not be the best approach for equations that are already in a factored form.

Q: How do I know if I have completed the square correctly?

A: You can check if you have completed the square correctly by plugging the solution back into the original equation. If the solution satisfies the equation, then you have completed the square correctly.

Q: What are some common mistakes to avoid when completing the square?

A: Some common mistakes to avoid when completing the square include:

• Not adding and subtracting the square of half the coefficient of $x$ correctly.
• Not creating a perfect square trinomial.
• Not simplifying the equation correctly.
• Not taking the square root of both sides to solve for $x$.

Q: Can I use completing the square to solve quadratic equations with complex coefficients?

A: Yes, completing the square can be used to solve quadratic equations with complex coefficients. However, you will need to use complex numbers and follow the same steps as before.

Q: How do I apply completing the square to solve quadratic equations with rational coefficients?

A: To apply completing the square to solve quadratic equations with rational coefficients, you will need to follow the same steps as before, but you will need to use rational numbers instead of complex numbers.

Conclusion

Completing the square is a powerful technique for solving quadratic equations. By following the steps outlined in this article, you can master this method and apply it to a wide range of problems. Remember to always start with the given equation, add and subtract the square of half the coefficient of $x$, create a perfect square trinomial, simplify the equation, and finally, take the square root of both sides to solve for $x$. With practice and patience, you will become proficient in completing the square and solving quadratic equations with ease.