Complete the square to solve the following quadratic equation: x2+2x−8=0.(1 point)

Question

Responses

x=−2, x=4
x equals negative 2 , x equals 4

x=−10, x=8
x equals negative 10 , x equals 8

x=2, x=−4
x equals 2 , x equals negative 4

x=10, x=−8

Answer 1

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

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

2. Complete the square: Take the coefficient of \(x\) (which is 2), divide it by 2 to get 1, and then square it to get 1. Add this square to both sides of the equation: \[ x^2 + 2x + 1 = 8 + 1 \] \[ (x + 1)^2 = 9 \]

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

Thus, the solutions to the equation \(x^2 + 2x - 8 = 0\) are: \[ x = 2, \quad x = -4 \]

So the correct response is: x=2, x=−4.

Answer 2

Complete the square to solve the following quadratic equation: x2−22x+1=15.(1 point)
Responses

x=−11+14−−√, −11−14−−√
x equals negative 11 plus Start Root 14 End Root comma negative 11 minus Start Root 14 End Root

x=11+315−−√, 11−315−−√
x equals 11 plus 3 Start Root 15 End Root comma 11 minus 3 Start Root 15 End Root

x=−11+315−−√, −11−315−−√
x equals negative 11 plus 3 Start Root 15 End Root comma negative 11 minus 3 Start Root 15 End Root

x=11+14−−√, 11−14−−√

Answer 3

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

1. Move the constant term to the other side: \[ x^2 - 22x = 15 - 1 \] \[ x^2 - 22x = 14 \]

2. Complete the square: Take the coefficient of \(x\) (which is \(-22\)), divide it by 2 to get \(-11\), and then square it to get \(121\). Add this square to both sides of the equation: \[ x^2 - 22x + 121 = 14 + 121 \] \[ (x - 11)^2 = 135 \]

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

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

Now, note that \(\sqrt{135} = \sqrt{9 \times 15} = 3\sqrt{15}\). Thus, the equation can be rewritten as: \[ x = 11 + 3\sqrt{15}, \quad x = 11 - 3\sqrt{15} \]

So the correct response is: x=11+3√15, 11−3√15.

Answer 4

Using the quadratic formula, find the solution to x2−6x−16=0.(1 point)
Responses

x=8, x=−8
x equals 8 , x equals negative 8

x=8, x=−2
x equals 8 , x equals negative 2

x=8
x equals 8

x=−2

Answer 5

To solve the quadratic equation \(x^2 - 6x - 16 = 0\) using the quadratic formula, we can use the following formula:

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

In this case, the coefficients are:

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

Now, plug these values into the formula:

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

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

3. This gives us two solutions:

   • 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: \[ x = 8, \quad x = -2 \]

So the correct response is: x=8, x=−2.

Answer 6

Which of the following is a solution to the quadratic equation 2x2−5x=6? Assume that the solution has been rounded to the nearest hundredth, if applicable.(1 point)
Responses

x=1.03
x equals 1.03

x=−0.89
x equals negative 0.89

x=0.89
x equals 0.89

x=−1.03

Answer 7

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

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

Now we'll use the quadratic formula:

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

In this case, the coefficients are:

• \(a = 2\)
• \(b = -5\)
• \(c = -6\)

Now, plug these values into the formula:

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

2. Now apply the values into the quadratic formula: \[ x = \frac{{-(-5) \pm \sqrt{73}}}{2 \cdot 2} \] \[ x = \frac{{5 \pm \sqrt{73}}}{4} \]

3. Calculate the two possible solutions:

   • First, calculate \(\sqrt{73} \approx 8.544\):
     • First solution: \[ x_1 = \frac{{5 + 8.544}}{4} \approx \frac{13.544}{4} \approx 3.386 \]
     • Second solution: \[ x_2 = \frac{{5 - 8.544}}{4} \approx \frac{-3.544}{4} \approx -0.886 \]

Now let's round \(x_2\) to the nearest hundredth: \[ x_2 \approx -0.89 \]

Therefore, the solutions to the equation \(2x^2 - 5x = 6\) (rounded to the nearest hundredth) are approximately:

• \(x \approx 3.39\)
• \(x \approx -0.89\)

So the correct response is: x=−0.89.

Answer 8

How many real solutions does the following quadratic equation have?

4x2+x+3=0

(1 point)
Responses

two real solutions
two real solutions

one real solution
one real solution

no real solutions
no real solutions

three real solutions

Answer 9

To determine how many real solutions the quadratic equation \(4x^2 + x + 3 = 0\) has, we can use the discriminant from the quadratic formula. The discriminant \(D\) is given by:

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

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

• \(a = 4\)
• \(b = 1\)
• \(c = 3\)

Now, let's calculate the discriminant:

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

Since the discriminant \(D\) is negative (\(-47\)), this indicates that the quadratic equation has no real solutions.

Thus, the correct response is: no real solutions.