Complete The Square For $x^2 + 8x - 3 = 0$.What Are The Solutions Using The Completing-the-square Method?A. $x = 4 \pm \sqrt{3}$B. $x = 4 \pm \sqrt{3}$C. $x = 4 \pm \sqrt{79}$D. $x = 4 \pm \sqrt{79}$

Feb 26, 2025 by ADMIN 200 views

**Completing the Square Method for Quadratic Equations**

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Introduction

Completing the square is a powerful method used to solve quadratic equations of the form $ax^2 + bx + c = 0$ . This method involves manipulating the quadratic equation to express it in the form $(x + p)^2 = q$ , where $p$ and $q$ are constants. In this article, we will use the completing-the-square method to solve the quadratic equation $x^2 + 8x - 3 = 0$ .

Step 1: Write Down the Quadratic Equation

The given quadratic equation is $x^2 + 8x - 3 = 0$ . Our goal is to rewrite this equation in the form $(x + p)^2 = q$ .

Step 2: Move the Constant Term to the Right-Hand Side

To begin the completing-the-square process, we need to move the constant term to the right-hand side of the equation. This gives us:

$x^2 + 8x = 3$

Step 3: Add and Subtract the Square of Half the Coefficient of x

The coefficient of $x$ is $8$ . Half of this coefficient is $4$ , and the square of $4$ is $16$ . We add and subtract $16$ to the left-hand side of the equation:

$x^2 + 8x + 16 - 16 = 3$

Step 4: Factor the Perfect Square Trinomial

The left-hand side of the equation is now a perfect square trinomial, which can be factored as:

$(x + 4)^2 - 16 = 3$

Step 5: Add 16 to Both Sides of the Equation

To isolate the perfect square trinomial on the left-hand side, we add $16$ to both sides of the equation:

$(x + 4)^2 = 3 + 16$

Step 6: Simplify the Right-Hand Side

Simplifying the right-hand side of the equation gives us:

$(x + 4)^2 = 19$

Step 7: Take the Square Root of Both Sides

To solve for $x$ , we take the square root of both sides of the equation:

$x + 4 = \pm \sqrt{19}$

Step 8: Subtract 4 from Both Sides

Finally, we subtract $4$ from both sides of the equation to solve for $x$ :

$x = -4 \pm \sqrt{19}$

Conclusion

Using the completing-the-square method, we have solved the quadratic equation $x^2 + 8x - 3 = 0$ . The solutions to this equation are $x = -4 \pm \sqrt{19}$ .

Comparison with Answer Choices

Comparing our solutions with the answer choices, we see that:

A. $x = 4 \pm \sqrt{3}$ is incorrect.
B. $x = 4 \pm \sqrt{3}$ is incorrect.
C. $x = 4 \pm \sqrt{79}$ is incorrect.
D. $x = 4 \pm \sqrt{79}$ is incorrect.

Our solutions match none of the answer choices. However, we can see that the correct solutions are $x = -4 \pm \sqrt{19}$ .

Final Answer

The final answer is $\boxed{x = -4 \pm \sqrt{19}}$ .

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Q: What is completing the square method?

A: Completing the square method is a technique used to solve quadratic equations of the form $ax^2 + bx + c = 0$ . It involves manipulating the quadratic equation to express it in the form $(x + p)^2 = q$ , where $p$ and $q$ are constants.

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

A: You should use completing the square method when you are given a quadratic equation that cannot be easily factored or solved using the quadratic formula.

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

A: The steps involved in completing the square method are:

Write down the quadratic equation.
Move the constant term to the right-hand side.
Add and subtract the square of half the coefficient of $x$ .
Factor the perfect square trinomial.
Add the same value to both sides of the equation.
Take the square root of both sides.
Solve for $x$ .

Q: What is the difference between completing the square method and quadratic formula?

A: Completing the square method and quadratic formula are two different techniques used to solve quadratic equations. Completing the square method involves manipulating the quadratic equation to express it in the form $(x + p)^2 = q$ , while the quadratic formula involves using the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to solve for $x$ .

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

A: No, completing the square method can only be used to solve quadratic equations that can be expressed in the form $ax^2 + bx + c = 0$ . It cannot be used to solve quadratic equations that are not in this form.

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

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

Not moving the constant term to the right-hand side.
Not adding and subtracting the square of half the coefficient of $x$ .
Not factoring the perfect square trinomial correctly.
Not adding the same value to both sides of the equation.
Not taking the square root of both sides correctly.

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

A: Yes, completing the square method can be used to solve quadratic equations with complex coefficients. However, the process may be more complicated and may involve using complex numbers.

Q: How do I know if completing the square method is the best technique to use for a particular quadratic equation?

A: You should use completing the square method if the quadratic equation can be easily expressed in the form $(x + p)^2 = q$ , or if the quadratic equation has a simple factorization. Otherwise, you may want to use the quadratic formula or another technique.

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

A: Yes, completing the square method can be used to solve quadratic equations with rational coefficients. However, the process may be more complicated and may involve using rational numbers.

Q: What are some real-world applications of completing the square method?

A: Completing the square method has many real-world applications, including:

Solving quadratic equations that arise in physics and engineering.
Modeling population growth and decline.
Analyzing data and making predictions.
Solving optimization problems.

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

A: Yes, completing the square method can be used to solve quadratic equations with negative coefficients. However, the process may be more complicated and may involve using negative numbers.

Q: How do I know if completing the square method is the best technique to use for a particular quadratic equation with negative coefficients?

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

A: Yes, completing the square method can be used to solve quadratic equations with complex roots. However, the process may be more complicated and may involve using complex numbers.

Q: What are some common mistakes to avoid when using completing the square method to solve quadratic equations with complex roots?

A: Some common mistakes to avoid when using completing the square method to solve quadratic equations with complex roots include:

Not using complex numbers correctly.
Not simplifying the expression correctly.
Not using the correct formula for complex roots.

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

A: Yes, completing the square method can be used to solve quadratic equations with irrational coefficients. However, the process may be more complicated and may involve using irrational numbers.

Q: What are some common mistakes to avoid when using completing the square method to solve quadratic equations with irrational coefficients?

A: Some common mistakes to avoid when using completing the square method to solve quadratic equations with irrational coefficients include:

Not using irrational numbers correctly.
Not simplifying the expression correctly.
Not using the correct formula for irrational roots.

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

A: No, completing the square method can only be used to solve quadratic equations with one variable. If you have a quadratic equation with multiple variables, you may want to use a different technique, such as substitution or elimination.

Q: What are some common mistakes to avoid when using completing the square method to solve quadratic equations with multiple variables?

A: Some common mistakes to avoid when using completing the square method to solve quadratic equations with multiple variables include:

Not using the correct formula for multiple variables.
Not simplifying the expression correctly.
Not using the correct technique for multiple variables.

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

A: No, completing the square method can only be used to solve quadratic equations with numeric coefficients. If you have a quadratic equation with non-numeric coefficients, you may want to use a different technique, such as algebraic manipulation or numerical methods.

Q: What are some common mistakes to avoid when using completing the square method to solve quadratic equations with non-numeric coefficients?

A: Some common mistakes to avoid when using completing the square method to solve quadratic equations with non-numeric coefficients include:

Not using the correct formula for non-numeric coefficients.
Not simplifying the expression correctly.
Not using the correct technique for non-numeric coefficients.

Q: Can I use completing the square method to solve quadratic equations with coefficients that are functions of other variables?

A: Yes, completing the square method can be used to solve quadratic equations with coefficients that are functions of other variables. However, the process may be more complicated and may involve using functions and algebraic manipulation.

Q: What are some common mistakes to avoid when using completing the square method to solve quadratic equations with coefficients that are functions of other variables?

A: Some common mistakes to avoid when using completing the square method to solve quadratic equations with coefficients that are functions of other variables include:

Not using the correct formula for coefficients that are functions of other variables.
Not simplifying the expression correctly.
Not using the correct technique for coefficients that are functions of other variables.

Q: Can I use completing the square method to solve quadratic equations with coefficients that are matrices?

A: Yes, completing the square method can be used to solve quadratic equations with coefficients that are matrices. However, the process may be more complicated and may involve using matrix algebra and linear transformations.

Q: What are some common mistakes to avoid when using completing the square method to solve quadratic equations with coefficients that are matrices?

A: Some common mistakes to avoid when using completing the square method to solve quadratic equations with coefficients that are matrices include:

Not using the correct formula for coefficients that are matrices.
Not simplifying the expression correctly.
Not using the correct technique for coefficients that are matrices.

Q: Can I use completing the square method to solve quadratic equations with coefficients that are tensors?

A: Yes, completing the square method can be used to solve quadratic equations with coefficients that are tensors. However, the process may be more complicated and may involve using tensor algebra and differential geometry.

Q: What are some common mistakes to avoid when using completing the square method to solve quadratic equations with coefficients that are tensors?

A: Some common mistakes to avoid when using completing the square method to solve quadratic equations with coefficients that are tensors include:

Not using the correct formula for coefficients that are tensors.
Not simplifying the expression correctly.
Not using the correct technique for coefficients that are tensors.

Q: Can I use completing the square method to solve quadratic equations with coefficients that are differential operators?

A: Yes, completing the square method can be used to solve quadratic equations with coefficients that are differential operators. However, the process may be more complicated and may involve using differential equations and operator theory.