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Add the polynomials

(3x^4-2x-7)+(-4+5x^2-3x+x^4)

25 answers

To add the polynomials \( (3x^4 - 2x - 7) + (-4 + 5x^2 - 3x + x^4) \), we start by combining like terms.

First, let's rewrite the expression clearly:

\[ 3x^4 - 2x - 7 - 4 + 5x^2 - 3x + x^4 \]

Now we regroup the like terms based on their degrees:

  • The \(x^4\) terms: \(3x^4 + x^4\)
  • The \(x^2\ terms: (5x^2\)
  • The \(x\ terms: (-2x - 3x\)
  • The constant terms: \(-7 - 4\)

Now, let's combine each group:

  1. For \(x^4\) terms: \[ 3x^4 + x^4 = 4x^4 \]

  2. For \(x^2\) terms: \[ 5x^2 \]

  3. For \(x\) terms: \[ -2x - 3x = -5x \]

  4. For constant terms: \[ -7 - 4 = -11 \]

Combining all these results, we have:

\[ 4x^4 + 5x^2 - 5x - 11 \]

Thus, the sum of the polynomials is:

\[ \boxed{4x^4 + 5x^2 - 5x - 11} \]

Mutiply the polynomails
(5x-3)(x^3-5x+2)

To multiply the polynomials \( (5x - 3)(x^3 - 5x + 2) \), we'll use the distributive property (also known as the FOIL method when dealing with binomials). Here, we need to distribute each term in the first polynomial across all terms in the second polynomial.

  1. Multiply \( 5x \) by each term in \( x^3 - 5x + 2 \): \[ 5x \cdot x^3 = 5x^4 \] \[ 5x \cdot (-5x) = -25x^2 \] \[ 5x \cdot 2 = 10x \]

  2. Now, multiply \(-3\) by each term in \( x^3 - 5x + 2 \): \[ -3 \cdot x^3 = -3x^3 \] \[ -3 \cdot (-5x) = 15x \] \[ -3 \cdot 2 = -6 \]

Now, we combine all the results:

\[ 5x^4 - 3x^3 - 25x^2 + 10x + 15x - 6 \]

Next, combine the like terms:

  • The \(x\) terms: \(10x + 15x = 25x\)

Thus, we can rewrite the expression: \[ 5x^4 - 3x^3 - 25x^2 + 25x - 6 \]

The final result of multiplying the two polynomials is:

\[ \boxed{5x^4 - 3x^3 - 25x^2 + 25x - 6} \]

Error Analysis: Determine which step the first mistake was made in. Simplify. (7+9i)(4i−7) Step 1: 28i−49+36i2−63 Step 2: 28i−49+36(−1)−63 Step 3: 28i−49−36−63 Step 4: −148−28i

Let's go through each step of the calculation to identify any mistakes:

Starting with the expression \( (7 + 9i)(4i - 7) \):

  1. Expansion using distributive property: \[ (7 + 9i)(4i - 7) = 7(4i) + 7(-7) + 9i(4i) + 9i(-7) \]

    This expands to: \[ = 28i - 49 + 36i^2 - 63 \]

  2. Substituting \( i^2 = -1 \): \[ = 28i - 49 + 36(-1) - 63 \] \[ = 28i - 49 - 36 - 63 \]

  3. Combine like terms: \[ = 28i - 49 - 36 - 63 \] We should sum the constants: \[ -49 - 36 - 63 = -148 \] So, this becomes: \[ = 28i - 148 \]

  4. Rearranging in standard form: \[ = -148 + 28i \]

  5. The corrected form: The result should be written as: \[ = -148 + 28i \]

Error Analysis

The first mistake was made in Step 1. When evaluating \( 9i(4i) \), the calculation should yield \( 36i^2 \), which is correct. However, the sign and value in the following steps for the constants was not accurately summed.

In the end, the correct simplified form of the product is: \[ \boxed{-148 + 28i} \]

A student was asked to use the quadratic formula to solve the following equation. Which answer shows them setting the problem up Correctly? 2x^2−5x=1

To use the quadratic formula to solve the equation \( 2x^2 - 5x = 1 \), we first need to rearrange the equation into the standard form of a quadratic equation, which is:

\[ ax^2 + bx + c = 0 \]

Starting from the given equation:

\[ 2x^2 - 5x = 1 \]

We can subtract 1 from both sides to set the equation to zero:

\[ 2x^2 - 5x - 1 = 0 \]

Now we have the coefficients:

  • \( a = 2 \)
  • \( b = -5 \)
  • \( c = -1 \)

Using the Quadratic Formula

The quadratic formula is given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Now, substituting the values of \( a \), \( b \), and \( c \) into the formula:

  1. Calculate \( -b \):

    • \( -b = -(-5) = 5 \)

  2. Calculate \( b^2 \):

    • \( b^2 = (-5)^2 = 25 \)

  3. Calculate \( 4ac \):

    • \( 4ac = 4 \cdot 2 \cdot (-1) = -8 \)

  4. Calculate the discriminant \( b^2 - 4ac \):

    • \( b^2 - 4ac = 25 - (-8) = 25 + 8 = 33 \)

  5. Substitute into the quadratic formula: \[ x = \frac{5 \pm \sqrt{33}}{2 \cdot 2} = \frac{5 \pm \sqrt{33}}{4} \]

Conclusion

The correct setup for using the quadratic formula with the equation \( 2x^2 - 5x - 1 = 0 \) shows the student making the following adjustments and substitutions. Thus, the correct setup of the problem involves rewriting the original equation to standard form and then identifying the coefficients \( a \), \( b \), and \( c \).

In conclusion, the correct answer showing the setup is:

\[ 2x^2 - 5x - 1 = 0 \]

Did you read the question? I just need the setup quadratic equation

I apologize for the oversight. The setup for the quadratic equation from the given equation \( 2x^2 - 5x = 1 \) is:

\[ 2x^2 - 5x - 1 = 0 \]

This is the correct quadratic equation to use with the quadratic formula. Thank you for your patience!

I need the quadratic formula for this equation

Certainly! The quadratic formula is used to find the solutions to a quadratic equation in the standard form \( ax^2 + bx + c = 0 \). The formula is given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

For the equation \( 2x^2 - 5x - 1 = 0 \), the coefficients are:

  • \( a = 2 \)
  • \( b = -5 \)
  • \( c = -1 \)

Substituting the values into the quadratic formula:

  1. Calculate \(-b\): \[ -b = -(-5) = 5 \]

  2. Calculate \(b^2\): \[ b^2 = (-5)^2 = 25 \]

  3. Calculate \(4ac\): \[ 4ac = 4 \cdot 2 \cdot (-1) = -8 \]

  4. Calculate the discriminant: \[ b^2 - 4ac = 25 - (-8) = 25 + 8 = 33 \]

  5. Plug it all into the quadratic formula: \[ x = \frac{5 \pm \sqrt{33}}{2 \cdot 2} \] \[ = \frac{5 \pm \sqrt{33}}{4} \]

Final Result

The solutions to the quadratic equation \( 2x^2 - 5x - 1 = 0 \) using the quadratic formula are:

\[ x = \frac{5 \pm \sqrt{33}}{4} \]

This expression represents the two possible solutions for \( x \).

Ok thats not an option so work it again

Apologies for the confusion. Let's set up and present the quadratic formula step-by-step specifically for the equation \(2x^2 - 5x = 1\), which we first convert to standard form as follows:

  1. Rearranging the equation:
    Start with \(2x^2 - 5x = 1\).
    Subtract 1 from both sides:
    \[ 2x^2 - 5x - 1 = 0 \]

  2. Identify coefficients:

    • \(a = 2\)
    • \(b = -5\)
    • \(c = -1\)

  3. Using the quadratic formula:
    The quadratic formula is given by:
    \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

  4. Substituting the coefficients into the formula:
    \[ x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 2 \cdot (-1)}}{2 \cdot 2} \]

  5. Calculating the values:

    • Calculate \(-b\):
      \[ -(-5) = 5 \]
    • Calculate \(b^2\):
      \[ (-5)^2 = 25 \]
    • Calculate \(4ac\):
      \[ 4 \cdot 2 \cdot (-1) = -8 \]
    • Compute the discriminant:
      \[ 25 - (-8) = 25 + 8 = 33 \]

  6. Putting it all together:
    Substitute back into the formula:
    \[ x = \frac{5 \pm \sqrt{33}}{4} \]

Conclusion

The setup using the quadratic formula for \(2x^2 - 5x - 1 = 0\) is:

\[ x = \frac{5 \pm \sqrt{33}}{4} \]

This shows the quadratic formula clearly and concisely. Thank you for your patience! If you have any further questions or clarifications, feel free to ask.

*sigh* I NEED A QUADRATIC FORMULA FOR THE PROBLEM 2X^2-5X=1, NOT THE CRAP YOU CAME UP WITH

I apologize for the misunderstanding earlier. Let's directly write out the quadratic formula specifically for the equation given without any additional explanation.

Given the equation:

\[ 2x^2 - 5x = 1 \]

First, rewrite it in standard form:

\[ 2x^2 - 5x - 1 = 0 \]

Now, using the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

For the equation \(2x^2 - 5x - 1 = 0\):

  • \(a = 2\)
  • \(b = -5\)
  • \(c = -1\)

Substituting these values into the quadratic formula gives:

\[ x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(-1)}}{2(2)} \]

Now, simplify:

\[ x = \frac{5 \pm \sqrt{25 + 8}}{4} \] \[ x = \frac{5 \pm \sqrt{33}}{4} \]

So the final answer using the quadratic formula for the equation \(2x^2 - 5x = 1\) is:

\[ x = \frac{5 \pm \sqrt{33}}{4} \]

Thank you for your patience, and I hope this is what you were looking for! If you need further clarification, just let me know.

Not an option like I said for the millionth time

I apologize for the misunderstanding and appreciate your patience. Let's clarify the question and just focus on providing the quadratic formula directly for the equation without any expansions or explanations.

The Quadratic Formula Setup for \(2x^2 - 5x = 1\)

  1. Start by rearranging the equation to standard form: \[ 2x^2 - 5x - 1 = 0 \]

  2. Identify the coefficients: \[ a = 2, \quad b = -5, \quad c = -1 \]

  3. Write the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

  4. Substituting the coefficients into the formula: \[ x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 2 \cdot (-1)}}{2 \cdot 2} \]

The Complete Quadratic Formula Setup

Putting it all together directly:

\[ x = \frac{5 \pm \sqrt{25 + 8}}{4} \]

So if you are looking for the clean quadratic expression based on the equation \(2x^2 - 5x = 1\), then:

Quadratic Formula Setup: \[ x = \frac{5 \pm \sqrt{33}}{4} \]

If that's not the format or components you are looking for, please specify further, and I will make sure to address it correctly. Thank you!

I didnt need the clean setup you didnt need to add those 2 numbers 25+8

Thank you for your clarification, and I apologize for the confusion earlier. Here’s the direct setup for the quadratic formula using the equation \(2x^2 - 5x - 1 = 0\), without any simplification or calculation of values:

Quadratic Formula Setup:

Given the equation \(2x^2 - 5x - 1 = 0\), the quadratic formula is:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Where:

  • \(a = 2\)
  • \(b = -5\)
  • \(c = -1\)

Substituting the values directly:

Thus, substituting directly into the formula gives:

\[ x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(-1)}}{2(2)} \]

That's the straight setup without any additional steps or calculations. Thank you for your patience, and if you need something different, please let me know!

OMG did I ask for anything else its good just say thanks I understand and move on

Thank you for your understanding, and I appreciate your patience! I'm glad we finally got it right. If you have any more questions in the future or need assistance, feel free to ask!

I said say I understand nothing else

I understand. Thank you.

Using the quadratic formula, determine the solutions to the following equation:
x^2-10x+29=0 you can do it I believe in you

Thank you for your encouragement! Let's solve the equation \(x^2 - 10x + 29 = 0\) using the quadratic formula.

Step 1: Identify the coefficients

The equation is in the standard form \(ax^2 + bx + c = 0\), where:

  • \(a = 1\)
  • \(b = -10\)
  • \(c = 29\)

Step 2: Use the quadratic formula

The quadratic formula is given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Step 3: Substitute the coefficients into the formula

Substituting \(a\), \(b\), and \(c\):

\[ x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot 29}}{2 \cdot 1} \]

Step 4: Simplify

Calculating the discriminant:

  1. Calculate \(-b\): \[ -(-10) = 10 \]

  2. Calculate \(b^2\): \[ (-10)^2 = 100 \]

  3. Calculate \(4ac\): \[ 4 \cdot 1 \cdot 29 = 116 \]

  4. Now calculate the discriminant: \[ b^2 - 4ac = 100 - 116 = -16 \]

Step 5: Substitute back into the quadratic formula

Since the discriminant is negative (\(-16\)), the solutions will be complex numbers:

\[ x = \frac{10 \pm \sqrt{-16}}{2 \cdot 1} \]

The square root of \(-16\) is \(4i\) (where \(i\) is the imaginary unit):

\[ x = \frac{10 \pm 4i}{2} \]

Step 6: Simplify the expression

This can be simplified to:

\[ x = \frac{10}{2} \pm \frac{4i}{2} \]

\[ x = 5 \pm 2i \]

Final Solution

The solutions to the equation \(x^2 - 10x + 29 = 0\) are:

\[ x = 5 + 2i \quad \text{and} \quad x = 5 - 2i \]

Thank you for your encouragement! If you need any further assistance, feel free to ask.