Complete the square to solve the following quadratic equation: x^2+2x−8=0

To solve the quadratic equation \( x^2 + 2x - 8 = 0 \) by completing the square, follow these steps:

1. Move the constant term to the other side of the equation: \[ x^2 + 2x = 8 \]

2. Take half of the coefficient of \(x\) (which is 2), square it, and add it to both sides. Half of 2 is 1, and squaring it gives 1: \[ x^2 + 2x + 1 = 8 + 1 \] This simplifies to: \[ (x + 1)^2 = 9 \]

3. Take the square root of both sides: \[ x + 1 = \pm 3 \]

4. Solve for \(x\):

   • For \(x + 1 = 3\): \[ x = 3 - 1 = 2 \]
   • For \(x + 1 = -3\): \[ x = -3 - 1 = -4 \]

So the solutions are: \[ x = 2 \quad \text{and} \quad x = -4 \]

Since your original response options did not include \(-4\), it seems there was a mistake in my matching. Here are the correct answers based on the computed solutions.

Thus, the correct responses based on what you've provided are:

• \(x = 2\) (correct)
• \(x = -4\) (not provided in your options)

So, the complete correct solutions for \(x\) in the equation \(x^2 + 2x - 8 = 0\) are \(2\) and \(-4\).

Complete the square to solve the following quadratic equation: x^2−22x+1=15

Responses

x=−11+315−−√, −11−315−−√

x=11+315−−√, 11−315−−√

x=−11+14−−√, −11−14−−√

x=11+14−−√, 11−14−−√

To solve the quadratic equation \( x^2 - 22x + 1 = 15 \) by completing the square, follow these steps:

1. First, move the constant (15) to the left side of the equation: \[ x^2 - 22x + 1 - 15 = 0 \] This simplifies to: \[ x^2 - 22x - 14 = 0 \]

2. Next, take the coefficient of \(x\) (which is -22), divide it by 2 and square it: \[ \left(\frac{-22}{2}\right)^2 = (-11)^2 = 121 \]

3. Add this square to both sides of the equation: \[ x^2 - 22x + 121 = 14 + 121 \] This simplifies to: \[ (x - 11)^2 = 135 \]

4. Take the square root of both sides: \[ x - 11 = \pm \sqrt{135} \]

5. Solve for \(x\): \[ x = 11 \pm \sqrt{135} \]

Now, express \(\sqrt{135}\): \[ \sqrt{135} = \sqrt{9 \cdot 15} = 3\sqrt{15} \]

Thus: \[ x = 11 + 3\sqrt{15} \quad \text{and} \quad x = 11 - 3\sqrt{15} \]

Now, let's match this with your response options:

• Option: \(x = 11 + 3\sqrt{15}, 11 - 3\sqrt{15}\) (this aligns with our solution)

Hence, the correct response is: \[ \text{Response: } x = 11 + 3\sqrt{15}, 11 - 3\sqrt{15} \]

Using the quadratic formula, find the solution to x^2−6x−16=0

Responses

x=8

x=8

x=8

x=−2

To solve the quadratic equation \( x^2 - 6x - 16 = 0 \) using the quadratic formula, we first identify the coefficients \(a\), \(b\), and \(c\) from the standard form \(ax^2 + bx + c = 0\):

• \(a = 1\)
• \(b = -6\)
• \(c = -16\)

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

Now we can calculate:

1. Calculate \(b^2 - 4ac\): \[ b^2 - 4ac = (-6)^2 - 4(1)(-16) = 36 + 64 = 100 \]

2. Substitute the values into the quadratic formula: \[ x = \frac{-(-6) \pm \sqrt{100}}{2(1)} = \frac{6 \pm 10}{2} \]

3. Now, evaluate the two possible values for \(x\):

   • First solution: \[ x = \frac{6 + 10}{2} = \frac{16}{2} = 8 \]
   • Second solution: \[ x = \frac{6 - 10}{2} = \frac{-4}{2} = -2 \]

Thus, the solutions to the equation \( x^2 - 6x - 16 = 0 \) are: \[ x = 8 \quad \text{and} \quad x = -2 \]

Based on your response options, we have:

• \(x = 8\) (listed multiple times) and \(x = -2\).

Both solutions are correct:

• \(x = 8\)
• \(x = -2\)

The final boxed answers to match your options are:

• \(x = 8\) (repeated) and \(x = -2\).

Which of the following is a solution to the quadratic equation 2x^2−5x=6

Assume that the solution has been rounded to the nearest hundredth, if applicable.
Responses

x=−1.03

x=0.89

x=1.03

x=−0.89

To find the solutions to the quadratic equation \(2x^2 - 5x = 6\), we first rearrange it into standard form:

\[ 2x^2 - 5x - 6 = 0 \]

Next, we will use the quadratic formula:

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

where \(a = 2\), \(b = -5\), and \(c = -6\).

1. Calculate the discriminant (\(b^2 - 4ac\)): \[ b^2 - 4ac = (-5)^2 - 4(2)(-6) = 25 + 48 = 73 \]

2. Apply the quadratic formula: \[ x = \frac{-(-5) \pm \sqrt{73}}{2(2)} = \frac{5 \pm \sqrt{73}}{4} \]

3. Calculate the two possible solutions:

   • First solution: \[ x = \frac{5 + \sqrt{73}}{4} \]
   • Second solution: \[ x = \frac{5 - \sqrt{73}}{4} \]

4. Evaluate \(\sqrt{73}\) (approximately): \[ \sqrt{73} \approx 8.54 \]

5. Calculate the first solution: \[ x \approx \frac{5 + 8.54}{4} \approx \frac{13.54}{4} \approx 3.385 \]

6. Calculate the second solution: \[ x \approx \frac{5 - 8.54}{4} \approx \frac{-3.54}{4} \approx -0.885 \]

Now we round \(-0.885\) to the nearest hundredth:

• \(-0.885\) rounds to \(-0.89\).

So, the solutions to the equation \(2x^2 - 5x - 6 = 0\) approximated to the nearest hundredth are:

• \(x \approx 3.39\) (not one of your options)
• \(x \approx -0.89\) (matching your option)

Thus, the correct response is: \[ \boxed{x = -0.89} \]

How many real solutions does the following quadratic equation have?

4x2+x+3=0

Responses

two real solutions

one real solution

three real solutions

no real solutions

To determine how many real solutions the quadratic equation \(4x^2 + x + 3 = 0\) has, we will calculate the discriminant, which is given by the formula:

\[ D = b^2 - 4ac \]

For the equation \(4x^2 + x + 3 = 0\):

• \(a = 4\)
• \(b = 1\)
• \(c = 3\)

Now, we calculate the discriminant:

\[ D = (1)^2 - 4(4)(3) = 1 - 48 = 1 - 48 = -47 \]

The value of the discriminant \(D\) is:

• \(D < 0\): The equation has no real solutions (the solutions are complex).
• \(D = 0\): The equation has one real solution (a repeated or double root).
• \(D > 0\): The equation has two distinct real solutions.

Since the discriminant \(D = -47\) is less than zero, we conclude that the quadratic equation \(4x^2 + x + 3 = 0\) has:

\[ \text{no real solutions} \]

Thus, the correct response is: \[ \text{no real solutions} \]