Question

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

Responses

x=10

x=−2

x=2
x=−10

Answer 1

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−−√

Answer 2

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

Responses

x=8

x=8

x=8

x=−2

Answer 3

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

Answer 4

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

Answer 5

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 \(-4\).

Answer 6

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:
\[
\text{Response: } x = 11 + 3\sqrt{15}, 11 - 3\sqrt{15}
\]

Answer 7

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 \(x = -2\).

Answer 8

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:
\[
\boxed{x = -0.89}
\]

Answer 9

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:
\[
\text{no real solutions}
\]