Cedric Completed The Square For The Quadratic Expression 5 X 2 − 40 X + 15 5x^2 - 40x + 15 5 X 2 − 40 X + 15 To Determine The Minimum Value Of The Expression, As Shown.Step 1: 5 ( X 2 − 8 X + 3 5(x^2 - 8x + 3 5 ( X 2 − 8 X + 3 ] Step 2: 5 ( X 2 − 8 X + 16 − 16 + 3 5(x^2 - 8x + 16 - 16 + 3 5 ( X 2 − 8 X + 16 − 16 + 3 ] Step 3: $5((x - 4)^2 -

Introduction

Completing the square is a powerful technique used to find the minimum or maximum value of a quadratic expression. It involves rewriting the quadratic expression in a form that allows us to easily identify the vertex of the parabola, which is the point where the expression reaches its minimum or maximum value. In this article, we will walk through the steps of completing the square for the quadratic expression $5x^2 - 40x + 15$ to determine its minimum value.

Step 1: Write the Quadratic Expression in General Form

The first step in completing the square is to write the quadratic expression in general form, which is $ax^2 + bx + c$. In this case, the quadratic expression is $5x^2 - 40x + 15$, so we can write it as:

$$5x^2 - 40x + 15$$

Step 2: Factor Out the Leading Coefficient

The next step is to factor out the leading coefficient, which is the coefficient of the $x^2$ term. In this case, the leading coefficient is 5, so we can factor it out as follows:

$$5(x^2 - 8x + 3)$$

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

The next step is to add and subtract the square of half the coefficient of the $x$ term. The coefficient of the $x$ term is -8, so half of it is -4. The square of -4 is 16, so we can add and subtract 16 as follows:

$$5(x^2 - 8x + 16 - 16 + 3)$$

Step 4: Rewrite the Expression as a Perfect Square

Now that we have added and subtracted 16, we can rewrite the expression as a perfect square. The perfect square is $(x - 4)^2$, so we can rewrite the expression as follows:

$$5((x - 4)^2 - 16 + 3)$$

Step 5: Simplify the Expression

The final step is to simplify the expression by combining like terms. We can simplify the expression as follows:

$$5(x - 4)^2 - 65$$

Discussion

Completing the square is a powerful technique used to find the minimum or maximum value of a quadratic expression. By following the steps outlined above, we can rewrite the quadratic expression in a form that allows us to easily identify the vertex of the parabola, which is the point where the expression reaches its minimum or maximum value.

In this case, the minimum value of the expression $5x^2 - 40x + 15$ is -65, which occurs when $x = 4$. This is because the expression is in the form $a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. In this case, the vertex is $(4, -65)$.

Conclusion

In conclusion, completing the square is a powerful technique used to find the minimum or maximum value of a quadratic expression. By following the steps outlined above, we can rewrite the quadratic expression in a form that allows us to easily identify the vertex of the parabola, which is the point where the expression reaches its minimum or maximum value.

Example Problems

Here are a few example problems that illustrate the concept of completing the square:

Example 1

Find the minimum value of the expression $x^2 - 6x + 2$.

Solution

To find the minimum value of the expression, we can complete the square as follows:

$$x^2 - 6x + 2 = (x - 3)^2 - 9 + 2 = (x - 3)^2 - 7$$

The minimum value of the expression is -7, which occurs when $x = 3$.

Example 2

Find the maximum value of the expression $x^2 + 4x - 3$.

Solution

To find the maximum value of the expression, we can complete the square as follows:

$$x^2 + 4x - 3 = (x + 2)^2 - 4 - 3 = (x + 2)^2 - 7$$

The maximum value of the expression is -7, which occurs when $x = -2$.

Tips and Tricks

Here are a few tips and tricks that can help you complete the square:

• Make sure to factor out the leading coefficient before adding and subtracting the square of half the coefficient of the $x$ term.
• Make sure to add and subtract the square of half the coefficient of the $x$ term, not just the coefficient itself.
• Make sure to rewrite the expression as a perfect square after adding and subtracting the square of half the coefficient of the $x$ term.
• Make sure to simplify the expression by combining like terms after rewriting it as a perfect square.

Common Mistakes

Here are a few common mistakes that can occur when completing the square:

• Failing to factor out the leading coefficient before adding and subtracting the square of half the coefficient of the $x$ term.
• Failing to add and subtract the square of half the coefficient of the $x$ term, not just the coefficient itself.
• Failing to rewrite the expression as a perfect square after adding and subtracting the square of half the coefficient of the $x$ term.
• Failing to simplify the expression by combining like terms after rewriting it as a perfect square.

Conclusion

Introduction

Completing the square is a powerful technique used to find the minimum or maximum value of a quadratic expression. In our previous article, we walked through the steps of completing the square for the quadratic expression $5x^2 - 40x + 15$ to determine its minimum value. 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 technique used to rewrite a quadratic expression in a form that allows us to easily identify the vertex of the parabola, which is the point where the expression reaches its minimum or maximum value.

Q: Why do we need to complete the square?

A: We need to complete the square because it allows us to easily identify the vertex of the parabola, which is the point where the expression reaches its minimum or maximum value. This is useful in a variety of applications, including optimization problems and graphing quadratic functions.

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

A: You should complete the square whenever you need to find the minimum or maximum value of a quadratic expression. This is typically the case when you are working with optimization problems or graphing quadratic functions.

Q: What are the steps to complete the square?

A: The steps to complete the square are as follows:

1. Write the quadratic expression in general form, which is $ax^2 + bx + c$.
2. Factor out the leading coefficient, which is the coefficient of the $x^2$ term.
3. Add and subtract the square of half the coefficient of the $x$ term.
4. Rewrite the expression as a perfect square.
5. Simplify the expression by combining like terms.

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

A: Completing the square and factoring are two different techniques used to rewrite quadratic expressions. Factoring involves expressing the quadratic expression as a product of two binomials, while completing the square involves rewriting the quadratic expression in a form that allows us to easily identify the vertex of the parabola.

Q: Can I use completing the square to factor quadratic expressions?

A: Yes, you can use completing the square to factor quadratic expressions. However, this is not always the most efficient method, and you may need to use other techniques, such as factoring or the quadratic formula, to factor the expression.

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

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

• Failing to factor out the leading coefficient before adding and subtracting the square of half the coefficient of the $x$ term.
• Failing to add and subtract the square of half the coefficient of the $x$ term, not just the coefficient itself.
• Failing to rewrite the expression as a perfect square after adding and subtracting the square of half the coefficient of the $x$ term.
• Failing to simplify the expression by combining like terms after rewriting it as a perfect square.

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 the expression back into the original equation and simplifying. If the expression simplifies to the original equation, then you have completed the square correctly.

Conclusion

In conclusion, completing the square is a powerful technique used to find the minimum or maximum value of a quadratic expression. By following the steps outlined above and avoiding common mistakes, you can use completing the square to rewrite quadratic expressions in a form that allows you to easily identify the vertex of the parabola.

Example Problems

Here are a few example problems that illustrate the concept of completing the square:

Example 1

Find the minimum value of the expression $x^2 - 6x + 2$.

Solution

To find the minimum value of the expression, we can complete the square as follows:

$$x^2 - 6x + 2 = (x - 3)^2 - 9 + 2 = (x - 3)^2 - 7$$

The minimum value of the expression is -7, which occurs when $x = 3$.

Example 2

Find the maximum value of the expression $x^2 + 4x - 3$.

Solution

To find the maximum value of the expression, we can complete the square as follows:

$$x^2 + 4x - 3 = (x + 2)^2 - 4 - 3 = (x + 2)^2 - 7$$

The maximum value of the expression is -7, which occurs when $x = -2$.

Tips and Tricks

Here are a few tips and tricks that can help you complete the square:

• Make sure to factor out the leading coefficient before adding and subtracting the square of half the coefficient of the $x$ term.
• Make sure to add and subtract the square of half the coefficient of the $x$ term, not just the coefficient itself.
• Make sure to rewrite the expression as a perfect square after adding and subtracting the square of half the coefficient of the $x$ term.
• Make sure to simplify the expression by combining like terms after rewriting it as a perfect square.

Common Mistakes

Here are a few common mistakes that can occur when completing the square:

• Failing to factor out the leading coefficient before adding and subtracting the square of half the coefficient of the $x$ term.
• Failing to add and subtract the square of half the coefficient of the $x$ term, not just the coefficient itself.
• Failing to rewrite the expression as a perfect square after adding and subtracting the square of half the coefficient of the $x$ term.
• Failing to simplify the expression by combining like terms after rewriting it as a perfect square.