Solve The Equation For All Values Of { X $}$ By Completing The Square.${ 2x^2 + 32x + 90 = 0 }$

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 quadratic equations by completing the square, a method that allows us to find the solutions of a quadratic equation in the form of $ax^2 + bx + c = 0$. We will use the given equation $2x^2 + 32x + 90 = 0$ as an example to demonstrate the steps involved in completing the square.

What is Completing the Square?

Completing the square is a technique used to solve quadratic equations by rewriting them in a perfect square trinomial form. This method involves manipulating the equation to create a perfect square trinomial on one side of the equation, which can then be factored into the square of a binomial. The resulting equation can be solved by setting each factor equal to zero and solving for the variable.

Step 1: Write the Equation in General Form

The first step in completing the square is to write the quadratic equation in general form, which is $ax^2 + bx + c = 0$. In this case, the equation is already in general form: $2x^2 + 32x + 90 = 0$.

Step 2: Move the Constant Term to the Right Side

The next step is to move the constant term to the right side of the equation by subtracting it from both sides. This gives us: $2x^2 + 32x = -90$.

Step 3: Divide by the Coefficient of the $x^2$ Term

To complete the square, we need to divide both sides of the equation by the coefficient of the $x^2$ term, which is 2. This gives us: $x^2 + 16x = -45$.

Step 4: Add and Subtract the Square of Half the Coefficient of the $x$ Term

The next step is to add and subtract the square of half the coefficient of the $x$ term to the left side of the equation. Half of 16 is 8, and the square of 8 is 64. Adding and subtracting 64 to the left side gives us: $x^2 + 16x + 64 - 64 = -45$.

Step 5: Factor the Perfect Square Trinomial

The left side of the equation is now a perfect square trinomial, which can be factored into the square of a binomial: $(x + 8)^2 - 64 = -45$.

Step 6: Simplify the Equation

The next step is to simplify the equation by combining like terms. This gives us: $(x + 8)^2 = 19$.

Step 7: Take the Square Root of Both Sides

To solve for $x$, we need to take the square root of both sides of the equation. This gives us: $x + 8 = \pm \sqrt{19}$.

Step 8: Solve for $x$

The final step is to solve for $x$ by subtracting 8 from both sides of the equation. This gives us: $x = -8 \pm \sqrt{19}$.

Conclusion

In this article, we have demonstrated how to solve a quadratic equation by completing the square. We have used the given equation $2x^2 + 32x + 90 = 0$ as an example to show the steps involved in completing the square. By following these steps, we have been able to find the solutions of the equation in the form of $x = -8 \pm \sqrt{19}$. This method is a powerful tool for solving quadratic equations and is an essential skill for students and professionals alike.

Example Problems

Problem 1

Solve the equation $x^2 + 6x + 8 = 0$ by completing the square.

Solution

To solve this equation, we need to follow the same steps as before. First, we move the constant term to the right side of the equation: $x^2 + 6x = -8$. Next, we divide both sides of the equation by the coefficient of the $x^2$ term, which is 1. This gives us: $x^2 + 6x = -8$. Then, we add and subtract the square of half the coefficient of the $x$ term to the left side of the equation. Half of 6 is 3, and the square of 3 is 9. Adding and subtracting 9 to the left side gives us: $x^2 + 6x + 9 - 9 = -8$. The left side of the equation is now a perfect square trinomial, which can be factored into the square of a binomial: $(x + 3)^2 - 9 = -8$. Simplifying the equation gives us: $(x + 3)^2 = 1$. Taking the square root of both sides of the equation gives us: $x + 3 = \pm 1$. Solving for $x$ gives us: $x = -3 \pm 1$. Therefore, the solutions of the equation are $x = -4$ and $x = -2$.

Problem 2

Solve the equation $3x^2 + 12x + 15 = 0$ by completing the square.

Solution

To solve this equation, we need to follow the same steps as before. First, we move the constant term to the right side of the equation: $3x^2 + 12x = -15$. Next, we divide both sides of the equation by the coefficient of the $x^2$ term, which is 3. This gives us: $x^2 + 4x = -5$. Then, we add and subtract the square of half the coefficient of the $x$ term to the left side of the equation. Half of 4 is 2, and the square of 2 is 4. Adding and subtracting 4 to the left side gives us: $x^2 + 4x + 4 - 4 = -5$. The left side of the equation is now a perfect square trinomial, which can be factored into the square of a binomial: $(x + 2)^2 - 4 = -5$. Simplifying the equation gives us: $(x + 2)^2 = 1$. Taking the square root of both sides of the equation gives us: $x + 2 = \pm 1$. Solving for $x$ gives us: $x = -2 \pm 1$. Therefore, the solutions of the equation are $x = -3$ and $x = -1$.

Final Thoughts

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Q: What is completing the square?

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

Q: How do I know if I can complete the square on a quadratic equation?

A: You can complete the square on a quadratic equation if the coefficient of the $x^2$ term is 1. If the coefficient of the $x^2$ term is not 1, you will need to divide both sides of the equation by the coefficient of the $x^2$ term before completing the square.

Q: What is the first step in completing the square?

A: The first step in completing the square is to write the quadratic equation in general form, which is $ax^2 + bx + c = 0$. Then, you need to move the constant term to the right side of the equation by subtracting it from both sides.

Q: How do I add and subtract the square of half the coefficient of the $x$ term?

A: To add and subtract the square of half the coefficient of the $x$ term, you need to find half of the coefficient of the $x$ term and then square it. For example, if the coefficient of the $x$ term is 6, you would find half of 6, which is 3, and then square it, which gives you 9. You would then add and subtract 9 to the left side of the equation.

Q: What is the final step in completing the square?

A: The final step in completing the square is to take the square root of both sides of the equation. This will give you the solutions of the equation in the form of $x = a \pm b$, where $a$ and $b$ are constants.

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

A: No, completing the square can only be used to solve quadratic equations that have a coefficient of 1 for the $x^2$ term. If the coefficient of the $x^2$ term is not 1, you will need to use a different method, such as factoring or the quadratic formula.

Q: Is completing the square a difficult method to learn?

A: Completing the square can be a bit challenging to learn at first, but with practice, it becomes easier. It's a good idea to start with simple quadratic equations and work your way up to more complex ones.

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

A: Yes, completing the square can be used to solve quadratic equations with complex solutions. However, you will need to use the imaginary unit $i$ to represent the complex solutions.

Q: Are there any other methods for solving quadratic equations besides completing the square?

A: Yes, there are several other methods for solving quadratic equations, including factoring, the quadratic formula, and graphing. Each method has its own advantages and disadvantages, and the choice of method will depend on the specific equation and the level of difficulty.

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

A: Yes, completing the square can be used to solve quadratic equations with rational coefficients. However, you will need to use the rational root theorem to find the possible rational solutions of the equation.

Q: Is completing the square a useful method for solving quadratic equations in real-world applications?

A: Yes, completing the square is a useful method for solving quadratic equations in real-world applications, such as physics, engineering, and economics. It can be used to model and solve problems involving quadratic equations, and it can provide valuable insights into the behavior of the solutions.

Q: Can I use completing the square to solve quadratic equations with multiple solutions?

A: Yes, completing the square can be used to solve quadratic equations with multiple solutions. However, you will need to use the quadratic formula to find the solutions, and you will need to consider the possibility of multiple solutions.

Q: Is completing the square a good method for solving quadratic equations with large coefficients?

A: No, completing the square is not a good method for solving quadratic equations with large coefficients. In such cases, it's better to use the quadratic formula or graphing to find the solutions.

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

A: Yes, completing the square can be used to solve quadratic equations with negative coefficients. However, you will need to use the quadratic formula to find the solutions, and you will need to consider the possibility of negative solutions.

Q: Is completing the square a useful method for solving quadratic equations in algebraic geometry?

A: Yes, completing the square is a useful method for solving quadratic equations in algebraic geometry. It can be used to find the solutions of quadratic equations in terms of algebraic curves and surfaces.

Q: Can I use completing the square to solve quadratic equations with non-integer coefficients?

A: Yes, completing the square can be used to solve quadratic equations with non-integer coefficients. However, you will need to use the quadratic formula to find the solutions, and you will need to consider the possibility of non-integer solutions.

Q: Is completing the square a good method for solving quadratic equations with complex coefficients?

A: No, completing the square is not a good method for solving quadratic equations with complex coefficients. In such cases, it's better to use the quadratic formula or graphing to find the solutions.

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

A: Yes, completing the square can be used to solve quadratic equations with irrational coefficients. However, you will need to use the quadratic formula to find the solutions, and you will need to consider the possibility of irrational solutions.

Q: Is completing the square a useful method for solving quadratic equations in number theory?

A: Yes, completing the square is a useful method for solving quadratic equations in number theory. It can be used to find the solutions of quadratic equations in terms of prime numbers and other number-theoretic concepts.

Q: Can I use completing the square to solve quadratic equations with coefficients that are not polynomials?

A: No, completing the square is not a good method for solving quadratic equations with coefficients that are not polynomials. In such cases, it's better to use the quadratic formula or graphing to find the solutions.

Q: Is completing the square a good method for solving quadratic equations with coefficients that are not rational?

A: No, completing the square is not a good method for solving quadratic equations with coefficients that are not rational. In such cases, it's better to use the quadratic formula or graphing to find the solutions.

Q: Can I use completing the square to solve quadratic equations with coefficients that are not real?

A: No, completing the square is not a good method for solving quadratic equations with coefficients that are not real. In such cases, it's better to use the quadratic formula or graphing to find the solutions.

Q: Is completing the square a useful method for solving quadratic equations in calculus?

A: Yes, completing the square is a useful method for solving quadratic equations in calculus. It can be used to find the solutions of quadratic equations in terms of derivatives and integrals.

Q: Can I use completing the square to solve quadratic equations with coefficients that are not functions?

A: No, completing the square is not a good method for solving quadratic equations with coefficients that are not functions. In such cases, it's better to use the quadratic formula or graphing to find the solutions.

Q: Is completing the square a good method for solving quadratic equations with coefficients that are not continuous?

A: No, completing the square is not a good method for solving quadratic equations with coefficients that are not continuous. In such cases, it's better to use the quadratic formula or graphing to find the solutions.

Q: Can I use completing the square to solve quadratic equations with coefficients that are not differentiable?

A: No, completing the square is not a good method for solving quadratic equations with coefficients that are not differentiable. In such cases, it's better to use the quadratic formula or graphing to find the solutions.

Q: Is completing the square a useful method for solving quadratic equations in differential equations?

A: Yes, completing the square is a useful method for solving quadratic equations in differential equations. It can be used to find the solutions of quadratic equations in terms of derivatives and integrals.

Q: Can I use completing the square to solve quadratic equations with coefficients that are not analytic?

A: No, completing the square is not a good method for solving quadratic equations with coefficients that are not analytic. In such cases, it's better to use the quadratic formula or graphing to find the solutions.

Q: Is completing the square a good method for solving quadratic equations with coefficients that are not meromorphic?

A: No, completing the square is not a good method for solving quadratic equations with coefficients that are not meromorphic. In such cases, it's better to use the quadratic formula or graphing to find the solutions.

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