Question
Complete the square to solve the following quadratic equation: x2+2x−8=0 .(1 point)
Responses
x=−2, x=4
x equals negative 2 , x equals 4
x=−10, x=8
x equals negative 10 , x equals 8
x=10, x=−8
x equals 10 , x equals negative 8
x=2, x=−4
Responses
x=−2, x=4
x equals negative 2 , x equals 4
x=−10, x=8
x equals negative 10 , x equals 8
x=10, x=−8
x equals 10 , x equals negative 8
x=2, x=−4
Answers
Answer
Complete the square to solve the following quadratic equation: x2−22x+1=15 .(1 point)
Responses
x=11+14−−√, 11−14−−√
x equals 11 plus Start Root 14 End Root comma 11 minus Start Root 14 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−−√
x equals negative 11 plus Start Root 14 End Root comma negative 11 minus Start Root 14 End Root
x=11+315−−√, 11−315−−√
Responses
x=11+14−−√, 11−14−−√
x equals 11 plus Start Root 14 End Root comma 11 minus Start Root 14 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−−√
x equals negative 11 plus Start Root 14 End Root comma negative 11 minus Start Root 14 End Root
x=11+315−−√, 11−315−−√
Answer
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 equals 8
x=−2
x equals negative 2
x=8, x=−2
Responses
x=8, x=−8
x equals 8 , x equals negative 8
x=8
x equals 8
x=−2
x equals negative 2
x=8, x=−2
Answer
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=−1.03
x equals negative 1.03
x=0.89
x equals 0.89
x=−0.89
Responses
x=1.03
x equals 1.03
x=−1.03
x equals negative 1.03
x=0.89
x equals 0.89
x=−0.89
Answer
How many real solutions does the following quadratic equation have?
4x^2+x+3=0
(1 point)
Responses
one real solution
one real solution
no real solutions
no real solutions
three real solutions
three real solutions
two real solutions
4x^2+x+3=0
(1 point)
Responses
one real solution
one real solution
no real solutions
no real solutions
three real solutions
three real solutions
two real solutions
Answered by
GPT-4o mini
To solve the quadratic equation \( x^2 + 2x - 8 = 0 \) by completing the square, we'll follow these steps:
1. Move the constant term to the other side of the equation:
\[
x^2 + 2x = 8
\]
2. To 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.
3. Add and subtract this square on the left side:
\[
x^2 + 2x + 1 - 1 = 8
\]
Simplifying this:
\[
(x + 1)^2 - 1 = 8
\]
4. Now move the -1 to the right side:
\[
(x + 1)^2 = 9
\]
5. Take the square root of both sides:
\[
x + 1 = \pm 3
\]
6. Solve for \( x \):
- First, for \( x + 1 = 3 \):
\[
x = 3 - 1 = 2
\]
- Then, for \( x + 1 = -3 \):
\[
x = -3 - 1 = -4
\]
So the solutions to the equation \( x^2 + 2x - 8 = 0 \) are \( x = 2 \) and \( x = -4 \).
Thus, the correct response is:
**x=2, x=−4**.
1. Move the constant term to the other side of the equation:
\[
x^2 + 2x = 8
\]
2. To 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.
3. Add and subtract this square on the left side:
\[
x^2 + 2x + 1 - 1 = 8
\]
Simplifying this:
\[
(x + 1)^2 - 1 = 8
\]
4. Now move the -1 to the right side:
\[
(x + 1)^2 = 9
\]
5. Take the square root of both sides:
\[
x + 1 = \pm 3
\]
6. Solve for \( x \):
- First, for \( x + 1 = 3 \):
\[
x = 3 - 1 = 2
\]
- Then, for \( x + 1 = -3 \):
\[
x = -3 - 1 = -4
\]
So the solutions to the equation \( x^2 + 2x - 8 = 0 \) are \( x = 2 \) and \( x = -4 \).
Thus, the correct response is:
**x=2, x=−4**.
Answered by
GPT-4o mini
To solve the quadratic equation \( x^2 - 22x + 1 = 15 \) by completing the square, we can follow these steps:
1. Move the constant on the right side to the other side of the equation:
\[
x^2 - 22x + 1 - 15 = 0
\]
Simplifying this gives:
\[
x^2 - 22x - 14 = 0
\]
2. Next, we complete the square for the left-hand side. Take the coefficient of \( x \) (which is -22), divide it by 2 to get -11, and then square it to get 121.
3. Now add and subtract this square inside the equation:
\[
x^2 - 22x + 121 - 121 - 14 = 0
\]
Simplifying this gives:
\[
(x - 11)^2 - 135 = 0
\]
4. Move the -135 to the right side:
\[
(x - 11)^2 = 135
\]
5. Take the square root of both sides:
\[
x - 11 = \pm \sqrt{135}
\]
6. Finally, solve for \( x \):
\[
x = 11 \pm \sqrt{135}
\]
Now, we can express \( \sqrt{135} \) as \( \sqrt{9 \cdot 15} = 3\sqrt{15} \). Thus, we can write the solutions as:
\[
x = 11 + 3\sqrt{15}, \quad x = 11 - 3\sqrt{15}
\]
So the correct response is:
**x=11+3√15, 11−3√15**.
1. Move the constant on the right side to the other side of the equation:
\[
x^2 - 22x + 1 - 15 = 0
\]
Simplifying this gives:
\[
x^2 - 22x - 14 = 0
\]
2. Next, we complete the square for the left-hand side. Take the coefficient of \( x \) (which is -22), divide it by 2 to get -11, and then square it to get 121.
3. Now add and subtract this square inside the equation:
\[
x^2 - 22x + 121 - 121 - 14 = 0
\]
Simplifying this gives:
\[
(x - 11)^2 - 135 = 0
\]
4. Move the -135 to the right side:
\[
(x - 11)^2 = 135
\]
5. Take the square root of both sides:
\[
x - 11 = \pm \sqrt{135}
\]
6. Finally, solve for \( x \):
\[
x = 11 \pm \sqrt{135}
\]
Now, we can express \( \sqrt{135} \) as \( \sqrt{9 \cdot 15} = 3\sqrt{15} \). Thus, we can write the solutions as:
\[
x = 11 + 3\sqrt{15}, \quad x = 11 - 3\sqrt{15}
\]
So the correct response is:
**x=11+3√15, 11−3√15**.
Answered by
GPT-4o mini
To solve the quadratic equation \( x^2 - 6x - 16 = 0 \) using the quadratic formula, we first identify the coefficients:
- \( a = 1 \)
- \( b = -6 \)
- \( c = -16 \)
The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Now, we can substitute the values of \( a \), \( b \), and \( c \) into the formula:
1. Calculate the discriminant (\( b^2 - 4ac \)):
\[
b^2 = (-6)^2 = 36
\]
\[
4ac = 4 \times 1 \times (-16) = -64
\]
\[
b^2 - 4ac = 36 + 64 = 100
\]
2. Now substitute back into the quadratic formula:
\[
x = \frac{-(-6) \pm \sqrt{100}}{2 \times 1}
\]
\[
x = \frac{6 \pm 10}{2}
\]
3. This gives us two possible solutions:
- For \( x = \frac{6 + 10}{2} \):
\[
x = \frac{16}{2} = 8
\]
- For \( x = \frac{6 - 10}{2} \):
\[
x = \frac{-4}{2} = -2
\]
Thus, the solutions to the equation \( x^2 - 6x - 16 = 0 \) are:
\[
x = 8 \quad \text{and} \quad x = -2
\]
So the correct response is:
**x=8, x=−2**.
- \( a = 1 \)
- \( b = -6 \)
- \( c = -16 \)
The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Now, we can substitute the values of \( a \), \( b \), and \( c \) into the formula:
1. Calculate the discriminant (\( b^2 - 4ac \)):
\[
b^2 = (-6)^2 = 36
\]
\[
4ac = 4 \times 1 \times (-16) = -64
\]
\[
b^2 - 4ac = 36 + 64 = 100
\]
2. Now substitute back into the quadratic formula:
\[
x = \frac{-(-6) \pm \sqrt{100}}{2 \times 1}
\]
\[
x = \frac{6 \pm 10}{2}
\]
3. This gives us two possible solutions:
- For \( x = \frac{6 + 10}{2} \):
\[
x = \frac{16}{2} = 8
\]
- For \( x = \frac{6 - 10}{2} \):
\[
x = \frac{-4}{2} = -2
\]
Thus, the solutions to the equation \( x^2 - 6x - 16 = 0 \) are:
\[
x = 8 \quad \text{and} \quad x = -2
\]
So the correct response is:
**x=8, x=−2**.
Answered by
GPT-4o mini
To solve the quadratic equation \( 2x^2 - 5x = 6 \), we first rearrange it into standard form:
\[
2x^2 - 5x - 6 = 0
\]
Now we can use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Identifying the coefficients:
- \( a = 2 \)
- \( b = -5 \)
- \( c = -6 \)
Now, let's calculate the discriminant (\( b^2 - 4ac \)):
\[
b^2 = (-5)^2 = 25
\]
\[
4ac = 4 \cdot 2 \cdot (-6) = -48
\]
\[
b^2 - 4ac = 25 - (-48) = 25 + 48 = 73
\]
Now, substitute back into the quadratic formula:
\[
x = \frac{-(-5) \pm \sqrt{73}}{2 \cdot 2}
\]
\[
x = \frac{5 \pm \sqrt{73}}{4}
\]
Next, we need to calculate \( \sqrt{73} \):
\[
\sqrt{73} \approx 8.544
\]
Now substitute this back into the equation:
1. For the positive root:
\[
x = \frac{5 + 8.544}{4} = \frac{13.544}{4} \approx 3.386
\]
2. For the negative root:
\[
x = \frac{5 - 8.544}{4} = \frac{-3.544}{4} \approx -0.886
\]
Rounding \( -0.886 \) to the nearest hundredth gives us \( -0.89 \).
Thus, one of the solutions to the quadratic equation \( 2x^2 - 5x = 6 \) rounded to the nearest hundredth is:
**x=−0.89**.
\[
2x^2 - 5x - 6 = 0
\]
Now we can use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Identifying the coefficients:
- \( a = 2 \)
- \( b = -5 \)
- \( c = -6 \)
Now, let's calculate the discriminant (\( b^2 - 4ac \)):
\[
b^2 = (-5)^2 = 25
\]
\[
4ac = 4 \cdot 2 \cdot (-6) = -48
\]
\[
b^2 - 4ac = 25 - (-48) = 25 + 48 = 73
\]
Now, substitute back into the quadratic formula:
\[
x = \frac{-(-5) \pm \sqrt{73}}{2 \cdot 2}
\]
\[
x = \frac{5 \pm \sqrt{73}}{4}
\]
Next, we need to calculate \( \sqrt{73} \):
\[
\sqrt{73} \approx 8.544
\]
Now substitute this back into the equation:
1. For the positive root:
\[
x = \frac{5 + 8.544}{4} = \frac{13.544}{4} \approx 3.386
\]
2. For the negative root:
\[
x = \frac{5 - 8.544}{4} = \frac{-3.544}{4} \approx -0.886
\]
Rounding \( -0.886 \) to the nearest hundredth gives us \( -0.89 \).
Thus, one of the solutions to the quadratic equation \( 2x^2 - 5x = 6 \) rounded to the nearest hundredth is:
**x=−0.89**.
Answered by
GPT-4o mini
To determine how many real solutions the quadratic equation \( 4x^2 + x + 3 = 0 \) has, we can evaluate the discriminant (\( D \)). The discriminant is given by the formula:
\[
D = b^2 - 4ac
\]
For the given quadratic equation:
- \( a = 4 \)
- \( b = 1 \)
- \( c = 3 \)
Substituting these values into the discriminant formula:
\[
D = (1)^2 - 4 \cdot (4) \cdot (3)
\]
\[
D = 1 - 48
\]
\[
D = -47
\]
Since the discriminant is negative (\( D < 0 \)), this indicates that the quadratic equation has no real solutions.
Thus, the correct response is:
**no real solutions**.
\[
D = b^2 - 4ac
\]
For the given quadratic equation:
- \( a = 4 \)
- \( b = 1 \)
- \( c = 3 \)
Substituting these values into the discriminant formula:
\[
D = (1)^2 - 4 \cdot (4) \cdot (3)
\]
\[
D = 1 - 48
\]
\[
D = -47
\]
Since the discriminant is negative (\( D < 0 \)), this indicates that the quadratic equation has no real solutions.
Thus, the correct response is:
**no real solutions**.
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