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use the quadratic formula to answer the questions below

quadratic formula is x = -b+ square b^2 - 4ac/ 2a
the equation is 2x^2 + 22x + 60
fill the blanks below ( yellow boxes )

part A
GCF
The gcf is =
the quadrtic with the factored out the gcf is =
A=
B=
c=
part B use the quadratic formula
x = ---- + ----^2 - 4(-----)(-----)/ 2 (-------)
part c solove for X
x = ------- (lesser number
x = -------- (greater number)

5 answers

Let's solve the given problem step by step.

Part A:

  1. Finding the GCF: The coefficients of the quadratic equation \(2x^2 + 22x + 60\) are:

    • \(2\) (coefficient of \(x^2\))
    • \(22\) (coefficient of \(x\))
    • \(60\) (constant term)

    The GCF of \(2\), \(22\), and \(60\) is \(2\).

  2. Factoring out the GCF: If we factor out \(2\), we get: \[ 2(x^2 + 11x + 30) \]

  3. Values for A, B, and C: The factored form yields the new coefficients:

    • \(A = 1\) (from \(x^2\))
    • \(B = 11\) (from \(11x\))
    • \(C = 30\) (the constant term)

So for Part A, we have:

  • The GCF is = 2
  • The quadratic with the factored out the GCF is = 2(x^2 + 11x + 30)
  • A = 1
  • B = 11
  • C = 30

Part B: Using the quadratic formula: \[ x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]

Plugging in the values we found:

  • A = 1
  • B = 11
  • C = 30

So, we can fill in: \[ x = \frac{-11 \pm \sqrt{11^2 - 4(1)(30)}}{2(1)} \]

Part C: Now, let's calculate the discriminant and solve for \(x\):

  1. Calculate \(B^2 - 4AC\): \[ B^2 - 4AC = 11^2 - 4 \cdot 1 \cdot 30 = 121 - 120 = 1 \]

  2. Now, plugging this into our formula: \[ x = \frac{-11 \pm \sqrt{1}}{2} \] \[ x = \frac{-11 \pm 1}{2} \]

  3. Calculating the two possible values for \(x\):

  • For the lesser number: \[ x = \frac{-11 - 1}{2} = \frac{-12}{2} = -6 \]
  • For the greater number: \[ x = \frac{-11 + 1}{2} = \frac{-10}{2} = -5 \]

So, we have:

  • x = -6 (lesser number)
  • x = -5 (greater number)

In summary, the filled blanks are:

  • The GCF is = 2

  • The quadratic with the factored out the GCF is = 2(x^2 + 11x + 30)

  • A = 1

  • B = 11

  • C = 30

  • x = -11 + sqrt{1} / 2

  • x = -6 (lesser number)

  • x = -5 (greater number)

part B is wrong rework it

Apologies for the confusion in Part B. Let's carefully apply the quadratic formula to the coefficients provided.

Given Quadratic Equation

The quadratic equation we are working with is: \[ 2x^2 + 22x + 60 \]

Step 1: Identify A, B, and C

From our analysis in Part A:

  • A = 2
  • B = 22
  • C = 60

Step 2: Write the Quadratic Formula

The quadratic formula is given by: \[ x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]

Part B: Substitute A, B, and C into the Formula

Substituting our values of A, B, and C into the quadratic formula: \[ x = \frac{-22 \pm \sqrt{22^2 - 4 \cdot 2 \cdot 60}}{2 \cdot 2} \]

Step 3: Calculate the Discriminant

Now, calculate the discriminant \(B^2 - 4AC\): \[ 22^2 = 484 \] \[ 4 \cdot 2 \cdot 60 = 480 \] Now, calculating the discriminant: \[ B^2 - 4AC = 484 - 480 = 4 \]

Step 4: Substitute the Discriminant Back into the Formula

Now plug the value of the discriminant back into the formula: \[ x = \frac{-22 \pm \sqrt{4}}{4} \]

Since \(\sqrt{4} = 2\): \[ x = \frac{-22 \pm 2}{4} \]

Part C: Solve for x

Calculating the two possible values:

  1. For the lesser number: \[ x = \frac{-22 - 2}{4} = \frac{-24}{4} = -6 \]
  2. For the greater number: \[ x = \frac{-22 + 2}{4} = \frac{-20}{4} = -5 \]

Final Answers for Part B and Part C:

  • \( x = -22 + 2 / 4 \)
  • x = -6 (lesser number)
  • x = -5 (greater number)

Thank you for your patience, and I hope this now accurately reflects your needs!

rework it again for the equation for part B

Certainly! Let’s set everything up correctly for Part B of the given quadratic equation \(2x^2 + 22x + 60\) using the quadratic formula properly without any mistakes.

Part A Recap

From previously discussed calculations, we identified:

  • A = 2
  • B = 22
  • C = 60

Part B: Using the Quadratic Formula

The correct quadratic formula is:

\[ x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]

Now substitute the values of A, B, and C into the quadratic formula:

  1. Writing the formula with the values substituted: \[ x = \frac{-22 \pm \sqrt{22^2 - 4(2)(60)}}{2(2)} \]

  2. Calculating the terms in the formula:

    • \(B^2 = 22^2 = 484\)
    • Calculate \(4AC\): \[ 4 \cdot 2 \cdot 60 = 480 \]
    • Calculating the discriminant: \[ B^2 - 4AC = 484 - 480 = 4 \]

  3. Substituting these values back into the equation: \[ x = \frac{-22 \pm \sqrt{4}}{4} \]

  4. Since \(\sqrt{4} = 2\): \[ x = \frac{-22 \pm 2}{4} \]

Part C: Solving for x

Now, we can solve for x by considering the two cases:

  1. For the lesser root: \[ x = \frac{-22 - 2}{4} = \frac{-24}{4} = -6 \]

  2. For the greater root: \[ x = \frac{-22 + 2}{4} = \frac{-20}{4} = -5 \]

Summary of Results

The correct entries for Part B are:

  • The quadratic equation with substituted values in the quadratic formula is: \[ x = \frac{-22 \pm \sqrt{22^2 - 4(2)(60)}}{2(2)} \]

Final Answers for Part C:

  • x = -6 (lesser number)
  • x = -5 (greater number)

Thank you for your patience, and I hope this now aligns perfectly with your request!