Question
25x2+60x+36=0
(1 point)
solution(s)
(1 point)
solution(s)
Answers
Answer
Complete the square to solve the following quadratic equation: x2+2x−8=0 .(1 point)
Responses
x=2, x=−4
x equals 2 , x equals negative 4
x=−2, x=4
x equals negative 2 , x equals 4
x=10, x=−8
x equals 10 , x equals negative 8
x=−10, x=8
Responses
x=2, x=−4
x equals 2 , x equals negative 4
x=−2, x=4
x equals negative 2 , x equals 4
x=10, x=−8
x equals 10 , x equals negative 8
x=−10, x=8
Answer
Complete the square to solve the following quadratic equation: x2−22x+1=15 .(1 point)
Responses
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+14−−√, 11−14−−√
x equals 11 plus Start Root 14 End Root comma 11 minus Start Root 14 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+315−−√, 11−315−−√
x equals 11 plus 3 Start Root 15 End Root comma 11 minus 3 Start Root 15 End Root
x=11+14−−√, 11−14−−√
x equals 11 plus Start Root 14 End Root comma 11 minus Start Root 14 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=−2
x equals 8 , x equals negative 2
x=8, x=−8
x equals 8 , x equals negative 8
x=8
x equals 8
x=−2
x equals negative 2
Responses
x=8, x=−2
x equals 8 , x equals negative 2
x=8, x=−8
x equals 8 , x equals negative 8
x=8
x equals 8
x=−2
x equals negative 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 negative 1.03
x=0.89
x equals 0.89
x=1.03
x equals 1.03
x=−0.89
Responses
x=−1.03
x equals negative 1.03
x=0.89
x equals 0.89
x=1.03
x equals 1.03
x=−0.89
Answer
How many real solutions does the following quadratic equation have?
4x2+x+3=0
(1 point)
Responses
one real solution
one real solution
two real solutions
two real solutions
no real solutions
no real solutions
three real solutions
4x2+x+3=0
(1 point)
Responses
one real solution
one real solution
two real solutions
two real solutions
no real solutions
no real solutions
three real solutions
Answered by
GPT-4o mini
To solve the quadratic equation \(25x^2 + 60x + 36 = 0\), we can use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \(a = 25\), \(b = 60\), and \(c = 36\).
1. Calculate the discriminant (\(b^2 - 4ac\)):
\[
b^2 = 60^2 = 3600
\]
\[
4ac = 4 \cdot 25 \cdot 36 = 3600
\]
\[
b^2 - 4ac = 3600 - 3600 = 0
\]
2. Since the discriminant is 0, there is one real solution (a repeated root).
3. Now substitute into the quadratic formula:
\[
x = \frac{-60 \pm \sqrt{0}}{2 \cdot 25} = \frac{-60}{50} = -\frac{6}{5} = -1.2
\]
Thus, the solution to the equation \(25x^2 + 60x + 36 = 0\) is:
\[
\boxed{-1.2}
\]
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \(a = 25\), \(b = 60\), and \(c = 36\).
1. Calculate the discriminant (\(b^2 - 4ac\)):
\[
b^2 = 60^2 = 3600
\]
\[
4ac = 4 \cdot 25 \cdot 36 = 3600
\]
\[
b^2 - 4ac = 3600 - 3600 = 0
\]
2. Since the discriminant is 0, there is one real solution (a repeated root).
3. Now substitute into the quadratic formula:
\[
x = \frac{-60 \pm \sqrt{0}}{2 \cdot 25} = \frac{-60}{50} = -\frac{6}{5} = -1.2
\]
Thus, the solution to the equation \(25x^2 + 60x + 36 = 0\) is:
\[
\boxed{-1.2}
\]
Answered by
GPT-4o mini
To solve the quadratic equation \(x^2 + 2x - 8 = 0\) by completing the square, follow these steps:
1. **Move the constant to the other side:**
\[
x^2 + 2x = 8
\]
2. **Complete the square:**
- Take half of the coefficient of \(x\), which is \(2\), half of \(2\) is \(1\), and then square it:
\[
1^2 = 1
\]
- Add \(1\) to both sides:
\[
x^2 + 2x + 1 = 8 + 1
\]
\[
(x + 1)^2 = 9
\]
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:
\[
x = 2 \quad \text{and} \quad x = -4
\]
Therefore, the correct response is:
**x = 2, x = -4**
1. **Move the constant to the other side:**
\[
x^2 + 2x = 8
\]
2. **Complete the square:**
- Take half of the coefficient of \(x\), which is \(2\), half of \(2\) is \(1\), and then square it:
\[
1^2 = 1
\]
- Add \(1\) to both sides:
\[
x^2 + 2x + 1 = 8 + 1
\]
\[
(x + 1)^2 = 9
\]
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:
\[
x = 2 \quad \text{and} \quad x = -4
\]
Therefore, 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, follow these steps:
1. **Rearrange the equation:**
Move the constant \(15\) to the left side:
\[
x^2 - 22x + 1 - 15 = 0
\]
This simplifies to:
\[
x^2 - 22x - 14 = 0
\]
2. **Complete the square:**
- Take half of the coefficient of \(x\) (which is \(-22\)), so half of \(-22\) is \(-11\), and then square it:
\[
(-11)^2 = 121
\]
- Add \(121\) to both sides:
\[
x^2 - 22x + 121 = 14 + 121
\]
- This results in:
\[
(x - 11)^2 = 135
\]
3. **Take the square root of both sides:**
\[
x - 11 = \pm \sqrt{135}
\]
4. **Solve for \(x\):**
\[
x = 11 \pm \sqrt{135}
\]
The square root of \(135\) can be simplified:
\[
\sqrt{135} = \sqrt{9 \times 15} = 3\sqrt{15}
\]
So, the final solutions are:
\[
x = 11 + 3\sqrt{15}, \quad x = 11 - 3\sqrt{15}
\]
Thus, the correct response is:
**x = 11 + 3√15, 11 - 3√15**
1. **Rearrange the equation:**
Move the constant \(15\) to the left side:
\[
x^2 - 22x + 1 - 15 = 0
\]
This simplifies to:
\[
x^2 - 22x - 14 = 0
\]
2. **Complete the square:**
- Take half of the coefficient of \(x\) (which is \(-22\)), so half of \(-22\) is \(-11\), and then square it:
\[
(-11)^2 = 121
\]
- Add \(121\) to both sides:
\[
x^2 - 22x + 121 = 14 + 121
\]
- This results in:
\[
(x - 11)^2 = 135
\]
3. **Take the square root of both sides:**
\[
x - 11 = \pm \sqrt{135}
\]
4. **Solve for \(x\):**
\[
x = 11 \pm \sqrt{135}
\]
The square root of \(135\) can be simplified:
\[
\sqrt{135} = \sqrt{9 \times 15} = 3\sqrt{15}
\]
So, the final solutions are:
\[
x = 11 + 3\sqrt{15}, \quad x = 11 - 3\sqrt{15}
\]
Thus, the correct response is:
**x = 11 + 3√15, 11 - 3√15**
Answered by
GPT-4o mini
To find the solutions to the equation \(x^2 - 6x - 16 = 0\) using the quadratic formula, we start with the formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \(a = 1\), \(b = -6\), and \(c = -16\).
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. **Substitute values into the quadratic formula**:
\[
x = \frac{-(-6) \pm \sqrt{100}}{2 \cdot 1}
\]
\[
x = \frac{6 \pm 10}{2}
\]
3. **Calculate the two possible values of \(x\)**:
- For the positive case:
\[
x = \frac{6 + 10}{2} = \frac{16}{2} = 8
\]
- For the negative case:
\[
x = \frac{6 - 10}{2} = \frac{-4}{2} = -2
\]
Thus, the solutions to the equation \(x^2 - 6x - 16 = 0\) are:
\[
x = 8 \quad \text{and} \quad x = -2
\]
Therefore, the correct response is:
**x = 8, x = -2**
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \(a = 1\), \(b = -6\), and \(c = -16\).
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. **Substitute values into the quadratic formula**:
\[
x = \frac{-(-6) \pm \sqrt{100}}{2 \cdot 1}
\]
\[
x = \frac{6 \pm 10}{2}
\]
3. **Calculate the two possible values of \(x\)**:
- For the positive case:
\[
x = \frac{6 + 10}{2} = \frac{16}{2} = 8
\]
- For the negative case:
\[
x = \frac{6 - 10}{2} = \frac{-4}{2} = -2
\]
Thus, the solutions to the equation \(x^2 - 6x - 16 = 0\) are:
\[
x = 8 \quad \text{and} \quad x = -2
\]
Therefore, the correct response is:
**x = 8, x = -2**
Answered by
GPT-4o mini
To find the solutions of the quadratic equation \(2x^2 - 5x = 6\), we first need to rearrange it into standard form:
\[
2x^2 - 5x - 6 = 0
\]
Next, we can use the quadratic formula, which is:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For our equation, \(a = 2\), \(b = -5\), and \(c = -6\).
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. **Substitute 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 values for \(x\)**:
- For the positive case:
\[
x_1 = \frac{5 + \sqrt{73}}{4}
\]
- For the negative case:
\[
x_2 = \frac{5 - \sqrt{73}}{4}
\]
4. **Approximate these solutions**:
First, calculate \(\sqrt{73}\):
\[
\sqrt{73} \approx 8.544
\]
Now find the two solutions:
- For \(x_1\):
\[
x_1 = \frac{5 + 8.544}{4} = \frac{13.544}{4} \approx 3.386
\]
- For \(x_2\):
\[
x_2 = \frac{5 - 8.544}{4} = \frac{-3.544}{4} \approx -0.886
\]
Rounding \(x_2\) to the nearest hundredth gives approximately \(-0.89\).
From the provided options, the solution rounded to the nearest hundredth is:
**x = -0.89**.
\[
2x^2 - 5x - 6 = 0
\]
Next, we can use the quadratic formula, which is:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For our equation, \(a = 2\), \(b = -5\), and \(c = -6\).
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. **Substitute 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 values for \(x\)**:
- For the positive case:
\[
x_1 = \frac{5 + \sqrt{73}}{4}
\]
- For the negative case:
\[
x_2 = \frac{5 - \sqrt{73}}{4}
\]
4. **Approximate these solutions**:
First, calculate \(\sqrt{73}\):
\[
\sqrt{73} \approx 8.544
\]
Now find the two solutions:
- For \(x_1\):
\[
x_1 = \frac{5 + 8.544}{4} = \frac{13.544}{4} \approx 3.386
\]
- For \(x_2\):
\[
x_2 = \frac{5 - 8.544}{4} = \frac{-3.544}{4} \approx -0.886
\]
Rounding \(x_2\) to the nearest hundredth gives approximately \(-0.89\).
From the provided options, the solution rounded to the nearest hundredth is:
**x = -0.89**.
Answered by
GPT-4o mini
To determine the number of real solutions for the quadratic equation \(4x^2 + x + 3 = 0\), we need to calculate the discriminant using the formula:
\[
D = b^2 - 4ac
\]
where \(a = 4\), \(b = 1\), and \(c = 3\).
1. **Calculate the discriminant**:
\[
D = (1)^2 - 4 \cdot (4) \cdot (3)
\]
\[
D = 1 - 48
\]
\[
D = -47
\]
2. **Analyze the discriminant**:
- If \(D > 0\), there are two distinct real solutions.
- If \(D = 0\), there is one real solution (a repeated root).
- If \(D < 0\), there are no real solutions.
Since \(D = -47\), which is less than 0, this means the quadratic equation has:
**no real solutions**.
Therefore, the correct response is:
**no real solutions**.
\[
D = b^2 - 4ac
\]
where \(a = 4\), \(b = 1\), and \(c = 3\).
1. **Calculate the discriminant**:
\[
D = (1)^2 - 4 \cdot (4) \cdot (3)
\]
\[
D = 1 - 48
\]
\[
D = -47
\]
2. **Analyze the discriminant**:
- If \(D > 0\), there are two distinct real solutions.
- If \(D = 0\), there is one real solution (a repeated root).
- If \(D < 0\), there are no real solutions.
Since \(D = -47\), which is less than 0, this means the quadratic equation has:
**no real solutions**.
Therefore, the correct response is:
**no real solutions**.
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