Solve The Quadratic Equation By Completing The Square. X 2 + 10 X + 15 = 0 X^2 + 10x + 15 = 0 X 2 + 10 X + 15 = 0 First, Choose The Appropriate Form And Fill In The Blanks With The Correct Numbers. Then, Solve The Equation. Simplify Your Answer As Much As Possible. If There Is More

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Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. 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 x2+10x+15=0x^2 + 10x + 15 = 0 by completing the square.

Choosing the Appropriate Form

Before we start solving the equation, we need to choose the appropriate form. The equation x2+10x+15=0x^2 + 10x + 15 = 0 is already in the standard form of a quadratic equation, which is ax2+bx+c=0ax^2 + bx + c = 0. However, to complete the square, we need to rewrite the equation in the form (x+p)2=q(x + p)^2 = q. To do this, we need to move the constant term to the right-hand side of the equation.

Step 1: Move the Constant Term

To move the constant term to the right-hand side, we subtract 15 from both sides of the equation:

x2+10x=−15x^2 + 10x = -15

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. The coefficient of x is 10, so half of it is 5. The square of 5 is 25. We add 25 to both sides of the equation:

x2+10x+25=−15+25x^2 + 10x + 25 = -15 + 25

Step 3: Simplify the Equation

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

(x+5)2=10(x + 5)^2 = 10

Solving the Equation

Now that we have the equation in the form (x+p)2=q(x + p)^2 = q, we can solve for x. To do this, we take the square root of both sides of the equation:

x+5=±10x + 5 = \pm \sqrt{10}

Step 4: Simplify the Square Root

The square root of 10 can be simplified as:

x+5=±10x + 5 = \pm \sqrt{10}

x+5=±3.16227766x + 5 = \pm 3.16227766

Step 5: Solve for x

Now, we can solve for x by subtracting 5 from both sides of the equation:

x=−5±3.16227766x = -5 \pm 3.16227766

Step 6: Simplify the Answer

Finally, we can simplify the answer by combining the two solutions:

x=−5+3.16227766x = -5 + 3.16227766 or x=−5−3.16227766x = -5 - 3.16227766

x=−1.83772234x = -1.83772234 or x=−8.16227766x = -8.16227766

Conclusion

In this article, we solved the quadratic equation x2+10x+15=0x^2 + 10x + 15 = 0 by completing the square. We started by choosing the appropriate form and filling in the blanks with the correct numbers. Then, we solved the equation by taking the square root of both sides and simplifying the answer. The final solutions were x=−1.83772234x = -1.83772234 or x=−8.16227766x = -8.16227766.

Tips and Tricks

  • When completing the square, make sure to add and subtract the square of half the coefficient of x.
  • When taking the square root of both sides of the equation, make sure to consider both the positive and negative square roots.
  • When simplifying the answer, make sure to combine like terms and simplify the square root.

Practice Problems

  • Solve the quadratic equation x2+6x+8=0x^2 + 6x + 8 = 0 by completing the square.
  • Solve the quadratic equation x2+2x+3=0x^2 + 2x + 3 = 0 by completing the square.
  • Solve the quadratic equation x2+4x+5=0x^2 + 4x + 5 = 0 by completing the square.

References

  • [1] "Quadratic Equations" by Math Open Reference
  • [2] "Completing the Square" by Purplemath
  • [3] "Quadratic Formula" by Khan Academy
    Quadratic Equations by Completing the Square: Q&A =====================================================

Introduction

In our previous article, we solved the quadratic equation x2+10x+15=0x^2 + 10x + 15 = 0 by completing the square. In this article, we will answer some frequently asked questions about quadratic equations and completing the square.

Q: What is completing the square?

A: Completing the square is a method used to solve quadratic equations by manipulating the equation to express it in a perfect square trinomial form. This form can then be easily solved.

Q: Why do we need to complete the square?

A: We need to complete the square because it allows us to solve quadratic equations that cannot be factored easily. By completing the square, we can express the equation in a form that can be easily solved.

Q: How do I know if I should complete the square?

A: You should complete the square if the quadratic equation cannot be factored easily. If the equation can be factored easily, you can use the factoring method to solve it.

Q: What are the steps to complete the square?

A: The steps to complete 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.
  3. Simplify the equation.
  4. Take the square root of both sides of the equation.
  5. Solve for x.

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

A: The quadratic formula is a method used to solve quadratic equations by plugging in the values of a, b, and c into a formula. Completing the square is a method used to solve quadratic equations by manipulating the equation to express it in a perfect square trinomial form.

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

A: No, you cannot use completing the square to solve all types of quadratic equations. Completing the square is only useful for quadratic equations that cannot be factored easily.

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

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

  • 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.
  • Not simplifying the equation.
  • Not taking the square root of both sides of the equation.
  • Not solving for x.

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 in the values of x into the original equation and checking if it is true.

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.

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

A: Some real-world applications of completing the square include:

  • Solving quadratic equations in physics and engineering.
  • Solving quadratic equations in economics and finance.
  • Solving quadratic equations in computer science and programming.

Conclusion

In this article, we answered some frequently asked questions about quadratic equations and completing the square. We hope that this article has been helpful in understanding the concept of completing the square and how to use it to solve quadratic equations.

Practice Problems

  • Solve the quadratic equation x2+6x+8=0x^2 + 6x + 8 = 0 by completing the square.
  • Solve the quadratic equation x2+2x+3=0x^2 + 2x + 3 = 0 by completing the square.
  • Solve the quadratic equation x2+4x+5=0x^2 + 4x + 5 = 0 by completing the square.

References

  • [1] "Quadratic Equations" by Math Open Reference
  • [2] "Completing the Square" by Purplemath
  • [3] "Quadratic Formula" by Khan Academy