Complete The Square To Rewrite The Quadratic Function In Vertex Form: Y = X 2 + 5 X + 4 Y = X^2 + 5x + 4 Y = X 2 + 5 X + 4 Answer Attempt 1 Out Of 2$y = $\square$ Submit Answer

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Introduction

Quadratic functions are a fundamental concept in mathematics, and understanding how to rewrite them in vertex form is crucial for solving various mathematical problems. In this article, we will focus on rewriting the quadratic function y=x2+5x+4y = x^2 + 5x + 4 in vertex form using the method of completing the square.

What is Completing the Square?

Completing the square is a mathematical technique used to rewrite a quadratic function in vertex form. It involves manipulating the quadratic expression to create a perfect square trinomial, which can be factored into the square of a binomial. This technique is useful for finding the vertex of a parabola, which is the point at which the parabola changes direction.

Step 1: Identify the Coefficients

To complete the square, we need to identify the coefficients of the quadratic function. In the given function y=x2+5x+4y = x^2 + 5x + 4, the coefficients are:

  • a=1a = 1 (coefficient of x2x^2)
  • b=5b = 5 (coefficient of xx)
  • c=4c = 4 (constant term)

Step 2: Calculate the Value to Complete the Square

To complete the square, we need to calculate the value that needs to be added to the quadratic expression to make it a perfect square trinomial. This value is given by the formula:

(b2)2=(52)2=254\left(\frac{b}{2}\right)^2 = \left(\frac{5}{2}\right)^2 = \frac{25}{4}

Step 3: Add and Subtract the Value

We add and subtract the calculated value to the quadratic expression:

y=x2+5x+254−254+4y = x^2 + 5x + \frac{25}{4} - \frac{25}{4} + 4

Step 4: Factor the Perfect Square Trinomial

The expression x2+5x+254x^2 + 5x + \frac{25}{4} is a perfect square trinomial, which can be factored into the square of a binomial:

x2+5x+254=(x+52)2x^2 + 5x + \frac{25}{4} = \left(x + \frac{5}{2}\right)^2

Step 5: Simplify the Expression

We simplify the expression by combining the constant terms:

y=(x+52)2−254+4y = \left(x + \frac{5}{2}\right)^2 - \frac{25}{4} + 4

Step 6: Write the Final Answer

The final answer is:

y=(x+52)2−94y = \left(x + \frac{5}{2}\right)^2 - \frac{9}{4}

Conclusion

In this article, we have shown how to rewrite the quadratic function y=x2+5x+4y = x^2 + 5x + 4 in vertex form using the method of completing the square. We have identified the coefficients, calculated the value to complete the square, added and subtracted the value, factored the perfect square trinomial, and simplified the expression. The final answer is y=(x+52)2−94y = \left(x + \frac{5}{2}\right)^2 - \frac{9}{4}.

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is given by:

y=a(x−h)2+ky = a(x - h)^2 + k

where (h,k)(h, k) is the vertex of the parabola. In our case, the vertex is (−52,−94)\left(-\frac{5}{2}, -\frac{9}{4}\right).

Example Problems

Here are some example problems to practice completing the square:

  1. Rewrite the quadratic function y=x2+6x+8y = x^2 + 6x + 8 in vertex form.
  2. Rewrite the quadratic function y=x2−4x+3y = x^2 - 4x + 3 in vertex form.
  3. Rewrite the quadratic function y=x2+2x+1y = x^2 + 2x + 1 in vertex form.

Solutions

  1. y=(x+3)2−5y = \left(x + 3\right)^2 - 5
  2. y=(x−2)2+1y = \left(x - 2\right)^2 + 1
  3. y=(x+1)2y = \left(x + 1\right)^2

Practice Problems

Here are some practice problems to help you master completing the square:

  1. Rewrite the quadratic function y=x2+7x+12y = x^2 + 7x + 12 in vertex form.
  2. Rewrite the quadratic function y=x2−3x+2y = x^2 - 3x + 2 in vertex form.
  3. Rewrite the quadratic function y=x2+4x+5y = x^2 + 4x + 5 in vertex form.

Answers

  1. y=(x+4)2−4y = \left(x + 4\right)^2 - 4
  2. y=(x−1)2+1y = \left(x - 1\right)^2 + 1
  3. y=(x+2)2−1y = \left(x + 2\right)^2 - 1

Conclusion

Introduction

Completing the square is a powerful technique used to rewrite quadratic functions in vertex form. In our previous article, we showed how to complete the square step-by-step. In this article, we will answer some frequently asked questions about completing the square.

Q: What is completing the square?

A: Completing the square is a mathematical technique used to rewrite a quadratic function in vertex form. It involves manipulating the quadratic expression to create a perfect square trinomial, which can be factored into the square of a binomial.

Q: Why is completing the square important?

A: Completing the square is important because it allows us to rewrite quadratic functions in vertex form, which is a more convenient and useful form for solving various mathematical problems. It also helps us to find the vertex of a parabola, which is the point at which the parabola changes direction.

Q: How do I know when to complete the square?

A: You should complete the square when you need to rewrite a quadratic function in vertex form. This is often the case when you are solving quadratic equations or graphing quadratic functions.

Q: What are the steps to complete the square?

A: The steps to complete the square are:

  1. Identify the coefficients of the quadratic function.
  2. Calculate the value to complete the square.
  3. Add and subtract the value to the quadratic expression.
  4. Factor the perfect square trinomial.
  5. Simplify the expression.

Q: What is the value to complete the square?

A: The value to complete the square is given by the formula:

(b2)2=(52)2=254\left(\frac{b}{2}\right)^2 = \left(\frac{5}{2}\right)^2 = \frac{25}{4}

Q: How do I factor the perfect square trinomial?

A: To factor the perfect square trinomial, you need to find the square root of the constant term and multiply it by the coefficient of the linear term.

Q: What is the vertex form of a quadratic function?

A: The vertex form of a quadratic function is given by:

y=a(x−h)2+ky = a(x - h)^2 + k

where (h,k)(h, k) is the vertex of the parabola.

Q: How do I find the vertex of a parabola?

A: To find the vertex of a parabola, you need to complete the square and rewrite the quadratic function in vertex form. The vertex is then given by the values of hh and kk in the vertex form.

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

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

  • Not identifying the coefficients correctly.
  • Not calculating the value to complete the square correctly.
  • Not adding and subtracting the value correctly.
  • Not factoring the perfect square trinomial correctly.
  • Not simplifying the expression correctly.

Q: How can I practice completing the square?

A: You can practice completing the square by working through example problems and exercises. You can also use online resources and practice tests to help you improve your skills.

Conclusion

In this article, we have answered some frequently asked questions about completing the square. We have covered topics such as the importance of completing the square, the steps to complete the square, and common mistakes to avoid. We hope that this article has been helpful in answering your questions and providing you with a better understanding of completing the square.