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Match the equation on the left to its equivalent on the right. (4 points)

left side

y = x^2 − 1x −20

y=(x+4)^2−5

y = x^2−6x+16

y=(x+8)(x+11)

right side

y = x^2 + 8x + 11

y = x^2 + 19x + 88

y = (x -3)^2 + 7

y = (x + 4) ( x - 5)

11 answers

To match the equations on the left with their equivalents on the right, we can simplify and manipulate each equation from the left side and see which right side form corresponds.

  1. Left side: \(y = x^2 - 1x - 20\)

    • Factor:
      \(y = x^2 - x - 20 = (x - 5)(x + 4)\)
    • Match: \(y = (x + 4)(x - 5)\) (This is from the right side)

  2. Left side: \(y = (x + 4)^2 - 5\)

    • Expand:
      \(y = (x^2 + 8x + 16) - 5 = x^2 + 8x + 11\)
    • Match: \(y = x^2 + 8x + 11\) (This is from the right side)

  3. Left side: \(y = x^2 - 6x + 16\)

    • Completing the square:
      \(y = (x^2 - 6x + 9) + 7 = (x - 3)^2 + 7\)
    • Match: \(y = (x - 3)^2 + 7\) (This is from the right side)

  4. Left side: \(y = x^2 + 19x + 88\)

    • This does not appear directly on the left side but implies some part of \(y = (x + 8)(x + 11)\) when set to zero. If we expand the right side:
      \((x + 8)(x + 11) = x^2 + 19x + 88\)
    • Match: \(y = (x + 8)(x + 11)\)

So the final matches are:

  1. \(y = x^2 - 1x - 20 \) ⇔ \(y = (x + 4)(x - 5)\)
  2. \(y = (x + 4)^2 - 5\) ⇔ \(y = x^2 + 8x + 11\)
  3. \(y = x^2 - 6x + 16\) ⇔ \(y = (x - 3)^2 + 7\)
  4. \(y = (x + 8)(x + 11)\) ⇔ \(y = x^2 + 19x + 88\)

The complete matched pairs are:

  • \(y = x^2 - 1x - 20 \) ↔ \(y = (x + 4)(x - 5)\)
  • \(y = (x + 4)^2 - 5\) ↔ \(y = x^2 + 8x + 11\)
  • \(y = x^2 - 6x + 16\) ↔ \(y = (x - 3)^2 + 7\)
  • \(y = (x + 8)(x + 11)\) ↔ \(y = x^2 + 19x + 88\)
You notice that the following problem cannot be factored so you solve it by completing the square.

What value of c would make the left-hand side of this equation a perfect square trinomial?

x^2 −14x + c = 13


(1 point)
Responses

4

-14

-7

196

49

To make the left-hand side of the equation \(x^2 - 14x + c\) a perfect square trinomial, we can use the completing the square method.

The general form of a perfect square trinomial is \((x - p)^2 = x^2 - 2px + p^2\).

In this case, we have:

  • The coefficient of \(x\) is \(-14\).
  • To find \(p\), we take \(-14\) and halve it, giving us \(-7\).
  • Now, to find \(c\), we square \(-7\): \[ c = (-7)^2 = 49 \]

Now we can check if this works: \[ x^2 - 14x + 49 = (x - 7)^2 \]

So the left-hand side becomes a perfect square trinomial when \(c = 49\).

Thus, the correct response is: 49.

Fill in the steps to complete the square:

x^2 +12x − 14 = 0

x^2 + 12x = 14

x^2 +12x +
Response area = 14+
Response area

We factor and simplify to get: (x+6)2 =
Response area

At this point, we need to Response area

After that is done, we will have:

Response area =
Response area

We get isolate the variable by:

Response area

Final answers are:

Response area

Let's fill in the steps to complete the square for the given equation \( x^2 + 12x - 14 = 0 \).

Step 1

Start with the original equation: \[ x^2 + 12x - 14 = 0 \]

Step 2

Move the constant to the other side: \[ x^2 + 12x = 14 \]

Step 3

To complete the square, we need to add a perfect square term. We take half of the coefficient of \(x\) (which is \(12\)), square it, and add it to both sides: \[ x^2 + 12x + (\frac{12}{2})^2 = 14 + (\frac{12}{2})^2 \]

Calculating \((\frac{12}{2})^2\): \[ (\frac{12}{2})^2 = 6^2 = 36 \]

So we fill in the response: \[ x^2 + 12x + 36 = 14 + 36 \]

Step 4

Now we factor the left-hand side: \[ (x + 6)^2 = 50 \]

Step 5

At this point, we need to take the square root of both sides: \[ x + 6 = \pm \sqrt{50} \]

Step 6

Simplifying \(\sqrt{50}\) gives: \[ \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2} \] So now we can write: \[ x + 6 = \pm 5\sqrt{2} \]

Step 7

Now, isolate the variable \(x\) by subtracting \(6\): \[ x = -6 \pm 5\sqrt{2} \]

Final Answer

The final answers are: \[ x = -6 + 5\sqrt{2} \quad \text{and} \quad x = -6 - 5\sqrt{2} \]

Here’s a summary of filled spaces:

  1. Response area: \(36\)
  2. Response area: \(50\)
  3. Response area: \(x + 6 = \pm \sqrt{50}\)
  4. Response area: \(x + 6 = \pm 5\sqrt{2}\)
  5. Response area: \(x = -6 \pm 5\sqrt{2}\)

So the completed steps are all filled out correctly!

there are eight total response areas that need answers

Thank you for your patience! Let's fill in all eight response areas step-by-step for the problem of completing the square for the equation \(x^2 + 12x - 14 = 0\).

Given:

\(x^2 + 12x - 14 = 0\)

Step 1:

Move the constant to the other side: \[ x^2 + 12x = 14 \]

Step 2:

To complete the square, take half of the coefficient of \(x\) (which is \(12\)), square it, and add it to both sides: \[ x^2 + 12x + \left(\frac{12}{2}\right)^2 = 14 + \left(\frac{12}{2}\right)^2 \] Calculating \(\left(\frac{12}{2}\right)^2\): \[ \left(\frac{12}{2}\right)^2 = 6^2 = 36 \]

Filling in Response Areas:

  1. Response area: \(36\) (the term to complete the square)
  2. Response area: \(14 + 36 = 50\) (the new right side after adding to both sides)

Step 3:

Now we factor the left-hand side: \[ x^2 + 12x + 36 = 50 \] which gives us: \[ (x + 6)^2 = 50 \]

  1. Response area: \( (x + 6)^2 \) (the factored left side)
  2. Response area: \(50\) (the right side)

Step 4:

Now we need to take the square root of both sides: \[ x + 6 = \pm \sqrt{50} \]

  1. Response area: \(x + 6 = \pm \sqrt{50}\) (after taking the square root)

Step 5:

Simplifying \(\sqrt{50}\): \[ \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2} \] So we write: \[ x + 6 = \pm 5\sqrt{2} \]

  1. Response area: \(x + 6 = \pm 5\sqrt{2}\) (after simplifying)

Step 6:

Now, isolate the variable \(x\) by subtracting \(6\): \[ x = -6 \pm 5\sqrt{2} \]

  1. Response area: \(x = -6 \pm 5\sqrt{2}\) (the final form)

Final Answers:

  1. Response area: \(x = -6 + 5\sqrt{2}\) and \(x = -6 - 5\sqrt{2}\) (the specific solutions)

Summary of All Response Areas:

  1. Response area: \(36\)
  2. Response area: \(50\)
  3. Response area: \((x + 6)^2\)
  4. Response area: \(50\)
  5. Response area: \(x + 6 = \pm \sqrt{50}\)
  6. Response area: \(x + 6 = \pm 5\sqrt{2}\)
  7. Response area: \(x = -6 \pm 5\sqrt{2}\)
  8. Response area: \(x = -6 + 5\sqrt{2}\) and \(x = -6 - 5\sqrt{2}\)

This should cover all eight response areas required!

When solving this quadratic by using the Quadratic Formula, what are the values of a, b, and c?

−3x2 +10x =−9


(3 points)

To identify the values of \(a\), \(b\), and \(c\) for the quadratic equation when using the Quadratic Formula, we need to arrange the equation in the standard form of a quadratic equation, which is:

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

Given the equation: \[ -3x^2 + 10x = -9 \]

Step 1: Rearrange to Standard Form

First, we need to move all terms to one side:

\[ -3x^2 + 10x + 9 = 0 \]

Step 2: Identify \(a\), \(b\), and \(c\)

Now, we can identify the coefficients:

  • \(a\) (coefficient of \(x^2\)): \(-3\)
  • \(b\) (coefficient of \(x\)): \(10\)
  • \(c\) (constant term): \(9\)

Final Values

So the values are:

  • \(a = -3\)
  • \(b = 10\)
  • \(c = 9\)
When solving this quadratic by using the Quadratic Formula, what are the values of a, b, and c?

−3x^2 +10x =−9


(3 points)

To find the values of \(a\), \(b\), and \(c\) from the quadratic equation given in the form \(−3x^2 + 10x = −9\), we first need to rearrange the equation into standard form \(ax^2 + bx + c = 0\).

Step 1: Rearrange to Standard Form

Starting with the equation: \[ -3x^2 + 10x = -9 \]

Add \(9\) to both sides to set the equation to zero: \[ -3x^2 + 10x + 9 = 0 \]

Step 2: Identify \(a\), \(b\), and \(c\)

Now we can identify the coefficients:

  • \(a\) is the coefficient of \(x^2\): \[ a = -3 \]

  • \(b\) is the coefficient of \(x\): \[ b = 10 \]

  • \(c\) is the constant term: \[ c = 9 \]

Final Values

Thus, the values are:

  • \(a = -3\)
  • \(b = 10\)
  • \(c = 9\)

These are the coefficients needed when applying the Quadratic Formula.