Solve the equation for all values of xx.

Question

2, x, left parenthesis, 2, x, plus, 9, right parenthesis, left parenthesis, 16, x, squared, minus, 9, right parenthesis, equals, 0
2x(2x+9)(16x
2
−9)=0

Answer 1

To solve the equation \(2x(2x + 9)(16x^2 - 9) = 0\), we can set each factor equal to zero and solve for \(x\).

1. First factor: \[ 2x = 0 \] Dividing both sides by 2 gives: \[ x = 0 \]

2. Second factor: \[ 2x + 9 = 0 \] Subtracting 9 from both sides: \[ 2x = -9 \] Dividing by 2 gives: \[ x = -\frac{9}{2} \]

3. Third factor: \[ 16x^2 - 9 = 0 \] Adding 9 to both sides gives: \[ 16x^2 = 9 \] Dividing by 16: \[ x^2 = \frac{9}{16} \] Taking the square root of both sides gives: \[ x = \frac{3}{4} \quad \text{or} \quad x = -\frac{3}{4} \]

Putting it all together, the solutions to the equation \(2x(2x + 9)(16x^2 - 9) = 0\) are:

\[ \boxed{0, -\frac{9}{2}, \frac{3}{4}, -\frac{3}{4}} \]

Answer 2

Solve the equation for all values of xx.
minus, x, left parenthesis, x, plus, 6, right parenthesis, left parenthesis, x, squared, minus, 49, right parenthesis, equals, 0
−x(x+6)(x
2
−49)=0

Answer 3

To solve the equation

\[ -x(x + 6)(x^2 - 49) = 0, \]

we can set each factor in the equation equal to zero:

1. First factor: \[ -x = 0 \] Dividing both sides by -1 gives: \[ x = 0. \]

2. Second factor: \[ x + 6 = 0 \] Subtracting 6 from both sides gives: \[ x = -6. \]

3. Third factor: \[ x^2 - 49 = 0 \] This can be factored as: \[ (x - 7)(x + 7) = 0. \] Setting each factor equal to zero gives: \[ x - 7 = 0 \quad \Rightarrow \quad x = 7, \] and \[ x + 7 = 0 \quad \Rightarrow \quad x = -7. \]

Putting it all together, the solutions to the equation \(-x(x + 6)(x^2 - 49) = 0\) are:

\[ \boxed{0, -6, 7, -7}. \]

Answer 4

Use the quadratic formula to solve. Express your answer in simplest form.
2, k, squared, plus, 11, k, plus, 12, equals, 0
2k
2
+11k+12=
0

Answer 5

To solve the quadratic equation

\[ 2k^2 + 11k + 12 = 0, \]

we will use the quadratic formula, which is given by:

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

where \(a\), \(b\), and \(c\) are the coefficients from the quadratic equation \(ax^2 + bx + c = 0\).

In this equation, we have:

• \(a = 2\),
• \(b = 11\),
• \(c = 12\).

Now we can calculate the discriminant:

\[ b^2 - 4ac = 11^2 - 4(2)(12) = 121 - 96 = 25. \]

Now substitute \(b\), \(\sqrt{b^2 - 4ac}\), and \(a\) into the quadratic formula:

\[ k = \frac{-11 \pm \sqrt{25}}{2(2)}. \]

Calculating \(\sqrt{25}\):

\[ \sqrt{25} = 5. \]

Now plug this into the formula:

\[ k = \frac{-11 \pm 5}{4}. \]

This gives us two possible solutions:

1. Using the plus sign: \[ k = \frac{-11 + 5}{4} = \frac{-6}{4} = -\frac{3}{2}. \]

2. Using the minus sign: \[ k = \frac{-11 - 5}{4} = \frac{-16}{4} = -4. \]

Thus, the solutions to the equation \(2k^2 + 11k + 12 = 0\) are:

\[ \boxed{-\frac{3}{2}, -4}. \]

Answer 6

Use the quadratic formula to solve. Express your answer in simplest form.
16, c, squared, minus, 24, c, plus, 4, equals, minus, 1
16c
2
−24c+4=
−1

Answer 7

To solve the equation

\[ 16c^2 - 24c + 4 = -1, \]

first, we rearrange the equation to set it to zero:

\[ 16c^2 - 24c + 4 + 1 = 0. \]

This simplifies to:

\[ 16c^2 - 24c + 5 = 0. \]

Now we can use the quadratic formula, which is given by:

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

where \(a\), \(b\), and \(c\) are the coefficients from the equation \(ax^2 + bx + c = 0\). In this case:

• \(a = 16\),
• \(b = -24\),
• \(c = 5\).

Next, we calculate the discriminant:

\[ b^2 - 4ac = (-24)^2 - 4(16)(5) = 576 - 320 = 256. \]

Now we can substitute \(b\), \(\sqrt{b^2 - 4ac}\), and \(a\) into the quadratic formula:

\[ c = \frac{-(-24) \pm \sqrt{256}}{2(16)}. \]

This simplifies to:

\[ c = \frac{24 \pm 16}{32}. \]

Now we can evaluate the two possible solutions:

1. Using the plus sign: \[ c = \frac{24 + 16}{32} = \frac{40}{32} = \frac{5}{4}. \]

2. Using the minus sign: \[ c = \frac{24 - 16}{32} = \frac{8}{32} = \frac{1}{4}. \]

Thus, the solutions to the equation \(16c^2 - 24c + 4 = -1\) are:

\[ \boxed{\frac{5}{4}, \frac{1}{4}}. \]

Answer 8

Graph the equation y, equals, minus, x, squared, plus, 14, x, minus, 48y=−x
2
+14x−48 on the accompanying set of axes. You must plot 5 points including the roots and the vertex. Using the graph, determine the equation of the axis of symmetry.
Click to plot points. Click points to delete them.

Answer 9

To graph the equation \( y = -x^2 + 14x - 48 \), we first need to analyze it by finding its roots and vertex which will allow us to plot the points accurately.

1. Finding the roots:

To find the roots, set \( y = 0 \):

\[ -x^2 + 14x - 48 = 0. \]

Multiplying by -1 gives:

\[ x^2 - 14x + 48 = 0. \]

Now, we can factor this quadratic equation. We are looking for two numbers that multiply to \( 48 \) and add to \( 14 \). These numbers are \( 6 \) and \( 8 \).

Thus, we can factor it as:

\[ (x - 6)(x - 8) = 0. \]

So the roots are:

\[ x = 6 \quad \text{and} \quad x = 8. \]

2. Finding the vertex:

The vertex \( x \)-coordinate can be found using the formula:

\[ x = -\frac{b}{2a}, \]

where \( a = -1 \) and \( b = 14 \):

\[ x = -\frac{14}{2 \cdot -1} = 7. \]

Now calculate the corresponding \( y \) value by substituting \( x = 7 \) back into the equation:

\[ y = -7^2 + 14 \cdot 7 - 48 = -49 + 98 - 48 = 1. \]

Thus, the vertex is at the point \( (7, 1) \).

3. Plotting the points:

We will now plot the following points:

• Roots: \( (6, 0) \) and \( (8, 0) \)
• Vertex: \( (7, 1) \)
• Additional points to get the shape of the graph. You can choose values around the vertex (for example \( x = 5 \) and \( x = 9 \)):
  • For \( x = 5 \): \[ y = -5^2 + 14 \cdot 5 - 48 = -25 + 70 - 48 = -3 , \Rightarrow \text{(5, -3)} \]
  • For \( x = 9 \): \[ y = -9^2 + 14 \cdot 9 - 48 = -81 + 126 - 48 = -3 , \Rightarrow \text{(9, -3)} \]

Now we have the points to plot:

• \( (6, 0) \)
• \( (8, 0) \)
• \( (7, 1) \)
• \( (5, -3) \)
• \( (9, -3) \)

4. Axis of symmetry:

The axis of symmetry for a parabola in the form \( y = ax^2 + bx + c \) is given by the equation \( x = h \), where \( h \) is the x-coordinate of the vertex. Since we found the vertex to be at \( (7, 1) \), the axis of symmetry is:

\[ \text{Axis of Symmetry: } x = 7. \]

Summary:

Plot Points:

• (6, 0)
• (8, 0)
• (7, 1)
• (5, -3)
• (9, -3)

Axis of Symmetry: \[ \boxed{x = 7} \]

You can now graph these points on a coordinate plane to visualize the parabola. The points should form a U-shaped curve opening downwards with the vertex at the highest point (7, 1) and passing through the roots at (6, 0) and (8, 0).

Answer 10

Answer type

Answer 11

It seems like you're looking for a structured response in the form of clear, labeled points and the equation for the axis of symmetry.

Key Points for the Graph of \( y = -x^2 + 14x - 48 \):

1. Roots (x-intercepts):

   • \( (6, 0) \)
   • \( (8, 0) \)

2. Vertex:

   • \( (7, 1) \)

3. Additional Points:

   • \( (5, -3) \)
   • \( (9, -3) \)

Summary of the Axis of Symmetry:

• The axis of symmetry is given by the vertical line that passes through the vertex.

\[ \text{Axis of Symmetry: } x = 7 \]

Final Presentation:

• Points to Plot:

  • \( (6, 0) \)
  • \( (8, 0) \)
  • \( (7, 1) \)
  • \( (5, -3) \)
  • \( (9, -3) \)

• Equation of the Axis of Symmetry: \[ \boxed{x = 7} \]

You can plot these points on a graph to visualize the parabolic curve. If you need further graphics or visual representation, please let me know!