How to Use Completing the Square to Solve Quadratic Equations

Oct 13, 2025 by talynt.org 98 views

Introduction

Completing the square is a powerful technique used to solve quadratic equations. It involves manipulating the equation to express it in a perfect square form, which can then be easily solved. In this article, we will use the completing the square method to solve the equation $x^2 - 8x = 3$ and find its solution set.

Step 1: Move the Constant Term to the Right Side

The first step in completing the square is to move the constant term to the right side of the equation. This gives us:

$x^2 - 8x - 3 = 0$

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

Next, we need to add and subtract the square of half the coefficient of $x$ to the left side of the equation. The coefficient of $x$ is $-8$ , so half of it is $-4$ . The square of $-4$ is $16$ . We add and subtract $16$ to the left side of the equation:

$x^2 - 8x + 16 - 16 - 3 = 0$

Step 3: Simplify the Equation

Now, we can simplify the equation by combining like terms:

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

$x^2 - 8x + 16 = 19$

Step 4: Factor the Left Side of the Equation

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

$(x - 4)^2 = 19$

Step 5: Take the Square Root of Both Sides

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

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

Step 6: Add 4 to Both Sides

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

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

Conclusion

The solution set of the equation $x^2 - 8x = 3$ is $\{4-\sqrt{19}, 4+\sqrt{19}\}$ . This is the set of all possible values of $x$ that satisfy the equation.

Answer

The correct answer is:

A. $\{4-\sqrt{19}, 4+\sqrt{19}\}$

Discussion

The completing the square method is a powerful technique used to solve quadratic equations. It involves manipulating the equation to express it in a perfect square form, which can then be easily solved. In this article, we used the completing the square method to solve the equation $x^2 - 8x = 3$ and found its solution set.

Why is Completing the Square Important?

Completing the square is an important technique in algebra because it allows us to solve quadratic equations in a straightforward and efficient manner. It is also a useful technique for solving equations that are not in the standard form $ax^2 + bx + c = 0$ . By completing the square, we can express the equation in a perfect square form, which can then be easily solved.

How to Use Completing the Square to Solve Quadratic Equations

To use completing the square to solve quadratic equations, we need to follow these steps:

Move the constant term to the right side of the equation.
Add and subtract the square of half the coefficient of $x$ to the left side of the equation.
Simplify the equation by combining like terms.
Factor the left side of the equation.
Take the square root of both sides of the equation.
Add the value that was added in step 2 to both sides of the equation.

Examples of Completing the Square

Here are some examples of completing the square:

$x^2 + 6x = 9$
$x^2 - 2x = 7$
$x^2 + 4x = 12$

Conclusion

Q: What is completing the square?

A: Completing the square is a technique used to solve quadratic equations by manipulating the equation to express it in a perfect square form.

Q: Why is completing the square important?

A: Completing the square is an important technique in algebra because it allows us to solve quadratic equations in a straightforward and efficient manner. It is also a useful technique for solving equations that are not in the standard form $ax^2 + bx + c = 0$ .

Q: How do I use completing the square to solve quadratic equations?

A: To use completing the square to solve quadratic equations, you need to follow these steps:

Move the constant term to the right side of the equation.
Add and subtract the square of half the coefficient of $x$ to the left side of the equation.
Simplify the equation by combining like terms.
Factor the left side of the equation.
Take the square root of both sides of the equation.
Add the value that was added in step 2 to both sides of the equation.

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

A: Completing the square and factoring are two different techniques used to solve quadratic equations. Factoring involves expressing the quadratic equation as a product of two binomials, while completing the square involves manipulating the equation to express it in a perfect square form.

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

A: No, completing the square can only be used to solve quadratic equations that can be expressed in the form $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants.

Q: What are some examples of quadratic equations that can be solved using completing the square?

A: Here are some examples of quadratic equations that can be solved using completing the square:

$x^2 + 6x = 9$
$x^2 - 2x = 7$
$x^2 + 4x = 12$

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

A: You should use completing the square when the quadratic equation is not easily factorable, or when you need to find the solutions of the equation in a specific form. You should use factoring when the quadratic equation can be easily expressed as a product of two binomials.

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

A: Yes, you can use completing the square to solve quadratic equations with complex coefficients. However, you need to be careful when taking the square root of complex numbers.

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

A: Here are some common mistakes to avoid when using completing the square:

Not moving the constant term to the right side of the equation.
Not adding and subtracting the square of half the coefficient of $x$ to the left side of the equation.
Not simplifying the equation by combining like terms.
Not factoring the left side of the equation.
Not taking the square root of both sides of the equation.
Not adding the value that was added in step 2 to both sides of the equation.

Conclusion

In conclusion, completing the square is a powerful technique used to solve quadratic equations. By following the steps outlined in this article, you can use completing the square to solve quadratic equations in a straightforward and efficient manner. Remember to avoid common mistakes and to use completing the square when the quadratic equation is not easily factorable.