Solve By Completing The Square:$m^2 - 14m + 39 = 0$Write Your Answers As Integers, Proper Or Improper Fractions Rounded To The Nearest Hundredth.$m =$ [ ] Or $m =$ [ ]

$$

Introduction

In algebra, solving quadratic equations is a crucial aspect of problem-solving. One of the methods used to solve quadratic equations is by completing the square. This method involves manipulating the equation to express it in a perfect square trinomial form, which can then be easily solved. In this article, we will focus on solving the quadratic equation $m^2 - 14m + 39 = 0$ by completing the square.

What is Completing the Square?

Completing the square is a method used to solve quadratic equations of the form $ax^2 + bx + c = 0$. The method involves manipulating the equation to express it in the form $(x + d)^2 = e$, where $d$ and $e$ are constants. This form can then be easily solved by taking the square root of both sides.

Step 1: Move the Constant Term to the Right-Hand Side

The first step in completing the square is to move the constant term to the right-hand side of the equation. In this case, we have:

$m^2 - 14m + 39 = 0$

We can rewrite this equation as:

$m^2 - 14m = -39$

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

The next step is to add and subtract the square of half the coefficient of $m$ to the left-hand side of the equation. The coefficient of $m$ is $-14$, so half of this is $-7$. The square of $-7$ is $49$. We can add and subtract $49$ to the left-hand side of the equation as follows:

$m^2 - 14m + 49 - 49 = -39$

Step 3: Express the Left-Hand Side as a Perfect Square Trinomial

The left-hand side of the equation can now be expressed as a perfect square trinomial:

$(m - 7)^2 - 49 = -39$

Step 4: Add $49$ to Both Sides of the Equation

To isolate the perfect square trinomial on the left-hand side, we can add $49$ to both sides of the equation:

$(m - 7)^2 = -39 + 49$

Step 5: Simplify the Right-Hand Side of the Equation

We can simplify the right-hand side of the equation as follows:

$(m - 7)^2 = 10$

Step 6: Take the Square Root of Both Sides of the Equation

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

$m - 7 = \pm \sqrt{10}$

Step 7: Add $7$ to Both Sides of the Equation

To isolate $m$, we can add $7$ to both sides of the equation:

$m = 7 \pm \sqrt{10}$

Conclusion

In this article, we have solved the quadratic equation $m^2 - 14m + 39 = 0$ by completing the square. The solution to the equation is $m = 7 \pm \sqrt{10}$. This solution can be expressed as a proper or improper fraction rounded to the nearest hundredth.

Final Answer

$m = 7 \pm \sqrt{10} \approx 7 \pm 3.16$

Therefore, the final answer is:

$m = 10.16$ or $m = 3.84$

Discussion

Completing the square is a powerful method used to solve quadratic equations. This method involves manipulating the equation to express it in a perfect square trinomial form, which can then be easily solved. In this article, we have used completing the square to solve the quadratic equation $m^2 - 14m + 39 = 0$. The solution to the equation is $m = 7 \pm \sqrt{10}$. This solution can be expressed as a proper or improper fraction rounded to the nearest hundredth.

Related Topics

• Solving quadratic equations by factoring
• Solving quadratic equations by using the quadratic formula
• Completing the square for quadratic equations in the form $ax^2 + bx + c = 0$

References

• [1] "Algebra" by Michael Artin
• [2] "College Algebra" by James Stewart
• [3] "Algebra and Trigonometry" by Michael Sullivan

Note: The references provided are for general information purposes only and are not specific to the topic of completing the square.

Introduction

Completing the square is a powerful method used to solve quadratic equations. In our previous article, we solved the quadratic equation $m^2 - 14m + 39 = 0$ by completing the square. In this article, we will answer some of the most frequently asked questions about completing the square.

Q: What is completing the square?

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

Q: How do I know if I should use completing the square to solve a quadratic equation?

A: You should use completing the square to solve a quadratic equation if the equation is in the form $ax^2 + bx + c = 0$ and you are not able to factor the equation or use the quadratic formula.

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

A: The steps involved in completing the square are:

1. Move the constant term to the right-hand side of the equation.
2. Add and subtract the square of half the coefficient of $x$ to the left-hand side of the equation.
3. Express the left-hand side as a perfect square trinomial.
4. Add the square of half the coefficient of $x$ to both sides of the equation.
5. Take the square root of both sides of the equation.
6. Add the coefficient of $x$ to both sides of the equation.

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

A: Factoring involves expressing a quadratic equation as a product of two binomials, while completing the square involves expressing a quadratic equation as a perfect square trinomial.

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 will need to use complex numbers to represent the solutions.

Q: How do I know if a quadratic equation can be solved by completing the square?

A: A quadratic equation can be solved by completing the square if the equation is in the form $ax^2 + bx + c = 0$ and the discriminant $b^2 - 4ac$ is a perfect square.

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

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

• Not moving the constant term to the right-hand side of the equation.
• Not adding and subtracting the square of half the coefficient of $x$ to the left-hand side of the equation.
• Not expressing the left-hand side as a perfect square trinomial.
• Not adding the square of half the coefficient of $x$ to both sides of the equation.
• Not taking the square root of both sides of the equation.

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

A: Yes, you can use completing the square to solve quadratic equations with rational coefficients. However, you will need to use rational numbers to represent the solutions.

Q: How do I know if a quadratic equation has real or complex solutions?

A: A quadratic equation has real solutions if the discriminant $b^2 - 4ac$ is a perfect square, and complex solutions if the discriminant is not a perfect square.

Conclusion

Completing the square is a powerful method used to solve quadratic equations. In this article, we have answered some of the most frequently asked questions about completing the square. We hope that this article has provided you with a better understanding of completing the square and how to use it to solve quadratic equations.

Final Answer

Completing the square is a method used to solve quadratic equations of the form $ax^2 + bx + c = 0$. The method involves manipulating the equation to express it in the form $(x + d)^2 = e$, where $d$ and $e$ are constants. By following the steps involved in completing the square, you can solve quadratic equations with real or complex coefficients.

Related Topics

• Solving quadratic equations by factoring
• Solving quadratic equations by using the quadratic formula
• Completing the square for quadratic equations in the form $ax^2 + bx + c = 0$

References

• [1] "Algebra" by Michael Artin
• [2] "College Algebra" by James Stewart
• [3] "Algebra and Trigonometry" by Michael Sullivan

Note: The references provided are for general information purposes only and are not specific to the topic of completing the square.