Question
Using the quadratic formula, find the solution to x2+6x+9=0
.(1 point)
.(1 point)
Answers
Answer
Solve −2x2+3x−1=−2
using the quadratic formula. Round the solution(s) to the nearest hundredths. Enter the smaller solution first.(1 point)
using the quadratic formula. Round the solution(s) to the nearest hundredths. Enter the smaller solution first.(1 point)
Answer
How many real solutions does the following quadratic equation have?
25x2+60x+36=0
25x2+60x+36=0
Answer
Question
Complete the square to solve the following quadratic equation: x2+2x−8=0
.(1 point)
Responses
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
x equals 2 , x equals negative 4
x=−2
, x=4
Complete the square to solve the following quadratic equation: x2+2x−8=0
.(1 point)
Responses
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
x equals 2 , x equals negative 4
x=−2
, x=4
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+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 negative 11 plus 3 Start Root 15 End Root comma negative 11 minus 3 Start Root 15 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+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 negative 11 plus 3 Start Root 15 End Root comma negative 11 minus 3 Start Root 15 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 equals 8
x=−2
x equals negative 2
x=8, x=−8
Responses
x=8, x=−2
x equals 8 , x equals negative 2
x=8
x equals 8
x=−2
x equals negative 2
x=8, x=−8
Answered by
GPT-4o mini
To find the solutions of the quadratic equation \( x^2 + 6x + 9 = 0 \) using the quadratic formula, we start by identifying the coefficients \( a \), \( b \), and \( c \):
- \( a = 1 \)
- \( b = 6 \)
- \( c = 9 \)
The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Now, we will calculate the discriminant \( b^2 - 4ac \):
\[
b^2 = 6^2 = 36
\]
\[
4ac = 4 \cdot 1 \cdot 9 = 36
\]
\[
b^2 - 4ac = 36 - 36 = 0
\]
Since the discriminant is \( 0 \), we will have one real root. Now substituting into the quadratic formula:
\[
x = \frac{-6 \pm \sqrt{0}}{2 \cdot 1}
\]
This simplifies to:
\[
x = \frac{-6 \pm 0}{2}
\]
Thus, we find:
\[
x = \frac{-6}{2} = -3
\]
Therefore, the solution to the equation \( x^2 + 6x + 9 = 0 \) is:
\[
\boxed{-3}
\]
- \( a = 1 \)
- \( b = 6 \)
- \( c = 9 \)
The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Now, we will calculate the discriminant \( b^2 - 4ac \):
\[
b^2 = 6^2 = 36
\]
\[
4ac = 4 \cdot 1 \cdot 9 = 36
\]
\[
b^2 - 4ac = 36 - 36 = 0
\]
Since the discriminant is \( 0 \), we will have one real root. Now substituting into the quadratic formula:
\[
x = \frac{-6 \pm \sqrt{0}}{2 \cdot 1}
\]
This simplifies to:
\[
x = \frac{-6 \pm 0}{2}
\]
Thus, we find:
\[
x = \frac{-6}{2} = -3
\]
Therefore, the solution to the equation \( x^2 + 6x + 9 = 0 \) is:
\[
\boxed{-3}
\]
Answered by
GPT-4o mini
To solve the equation \(-2x^2 + 3x - 1 = -2\), we first rearrange it into the standard form of a quadratic equation, \(ax^2 + bx + c = 0\).
Starting with the equation:
\[
-2x^2 + 3x - 1 + 2 = 0
\]
This simplifies to:
\[
-2x^2 + 3x + 1 = 0
\]
Now, we identify the coefficients:
- \(a = -2\)
- \(b = 3\)
- \(c = 1\)
Next, we apply the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
We first calculate the discriminant \(b^2 - 4ac\):
\[
b^2 = 3^2 = 9
\]
\[
4ac = 4 \cdot (-2) \cdot 1 = -8
\]
\[
b^2 - 4ac = 9 - (-8) = 9 + 8 = 17
\]
Now substituting back into the quadratic formula:
\[
x = \frac{-3 \pm \sqrt{17}}{2 \cdot -2}
\]
This becomes:
\[
x = \frac{-3 \pm \sqrt{17}}{-4}
\]
Now we separate the two possible solutions:
1.
\[
x_1 = \frac{-3 + \sqrt{17}}{-4}
\]
2.
\[
x_2 = \frac{-3 - \sqrt{17}}{-4}
\]
Calculating the numerical values for these solutions:
1. **Calculate \(\sqrt{17}\)**:
\(\sqrt{17} \approx 4.123\)
So:
\[
x_1 = \frac{-3 + 4.123}{-4} = \frac{1.123}{-4} \approx -0.28075 \text{ (rounding gives } -0.28 \text{ )}
\]
\[
x_2 = \frac{-3 - 4.123}{-4} = \frac{-7.123}{-4} \approx 1.78075 \text{ (rounding gives } 1.78 \text{ )}
\]
Thus the solutions, rounded to the nearest hundredths, are:
- Smaller solution: \(-0.28\)
- Larger solution: \(1.78\)
In conclusion, the solutions are:
\[
\boxed{-0.28} \quad \text{and} \quad \boxed{1.78}
\]
Starting with the equation:
\[
-2x^2 + 3x - 1 + 2 = 0
\]
This simplifies to:
\[
-2x^2 + 3x + 1 = 0
\]
Now, we identify the coefficients:
- \(a = -2\)
- \(b = 3\)
- \(c = 1\)
Next, we apply the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
We first calculate the discriminant \(b^2 - 4ac\):
\[
b^2 = 3^2 = 9
\]
\[
4ac = 4 \cdot (-2) \cdot 1 = -8
\]
\[
b^2 - 4ac = 9 - (-8) = 9 + 8 = 17
\]
Now substituting back into the quadratic formula:
\[
x = \frac{-3 \pm \sqrt{17}}{2 \cdot -2}
\]
This becomes:
\[
x = \frac{-3 \pm \sqrt{17}}{-4}
\]
Now we separate the two possible solutions:
1.
\[
x_1 = \frac{-3 + \sqrt{17}}{-4}
\]
2.
\[
x_2 = \frac{-3 - \sqrt{17}}{-4}
\]
Calculating the numerical values for these solutions:
1. **Calculate \(\sqrt{17}\)**:
\(\sqrt{17} \approx 4.123\)
So:
\[
x_1 = \frac{-3 + 4.123}{-4} = \frac{1.123}{-4} \approx -0.28075 \text{ (rounding gives } -0.28 \text{ )}
\]
\[
x_2 = \frac{-3 - 4.123}{-4} = \frac{-7.123}{-4} \approx 1.78075 \text{ (rounding gives } 1.78 \text{ )}
\]
Thus the solutions, rounded to the nearest hundredths, are:
- Smaller solution: \(-0.28\)
- Larger solution: \(1.78\)
In conclusion, the solutions are:
\[
\boxed{-0.28} \quad \text{and} \quad \boxed{1.78}
\]
Answered by
GPT-4o mini
To determine how many real solutions the quadratic equation \(25x^2 + 60x + 36 = 0\) has, we need to calculate the discriminant using the formula:
\[
D = b^2 - 4ac
\]
where \(a\), \(b\), and \(c\) are the coefficients from the standard form \(ax^2 + bx + c = 0\).
For the given equation:
- \(a = 25\)
- \(b = 60\)
- \(c = 36\)
Now, substituting these values into the discriminant formula:
\[
D = 60^2 - 4 \cdot 25 \cdot 36
\]
Calculating \(60^2\):
\[
60^2 = 3600
\]
Now calculating \(4 \cdot 25 \cdot 36\):
\[
4 \cdot 25 = 100
\]
\[
100 \cdot 36 = 3600
\]
Now putting it all together:
\[
D = 3600 - 3600 = 0
\]
The value of the discriminant \(D\) is \(0\).
### Conclusion:
When the discriminant is:
- \(D > 0\): There are two distinct real solutions.
- \(D = 0\): There is exactly one real solution (a double root).
- \(D < 0\): There are no real solutions.
Since \(D = 0\) for this equation, it has **exactly one real solution**.
\[
D = b^2 - 4ac
\]
where \(a\), \(b\), and \(c\) are the coefficients from the standard form \(ax^2 + bx + c = 0\).
For the given equation:
- \(a = 25\)
- \(b = 60\)
- \(c = 36\)
Now, substituting these values into the discriminant formula:
\[
D = 60^2 - 4 \cdot 25 \cdot 36
\]
Calculating \(60^2\):
\[
60^2 = 3600
\]
Now calculating \(4 \cdot 25 \cdot 36\):
\[
4 \cdot 25 = 100
\]
\[
100 \cdot 36 = 3600
\]
Now putting it all together:
\[
D = 3600 - 3600 = 0
\]
The value of the discriminant \(D\) is \(0\).
### Conclusion:
When the discriminant is:
- \(D > 0\): There are two distinct real solutions.
- \(D = 0\): There is exactly one real solution (a double root).
- \(D < 0\): There are no real solutions.
Since \(D = 0\) for this equation, it has **exactly one real solution**.
Answered by
GPT-4o mini
To solve the quadratic equation \( x^2 + 2x - 8 = 0 \) by completing the square, we will follow these steps:
1. **Move the constant term to the other side of the equation:**
\[
x^2 + 2x = 8
\]
2. **Complete the square:**
- Take half of the coefficient of \(x\), which is \(2\). Half of \(2\) is \(1\), and squaring it gives \(1^2 = 1\).
- Add and subtract this square (\(1\)) inside the equation.
\[
x^2 + 2x + 1 - 1 = 8
\]
This simplifies to:
\[
(x + 1)^2 - 1 = 8
\]
Now add \(1\) to both sides:
\[
(x + 1)^2 = 9
\]
3. **Take the square root of both sides:**
\[
x + 1 = \pm 3
\]
4. **Solve for \(x\):**
- First solution:
\[
x + 1 = 3 \implies x = 3 - 1 = 2
\]
- Second solution:
\[
x + 1 = -3 \implies x = -3 - 1 = -4
\]
Thus, the solutions are:
\[
x = 2 \quad \text{and} \quad x = -4
\]
Rounding off to the options given, the correct response is:
\[
\boxed{x = 2, \; x = -4}
\]
1. **Move the constant term to the other side of the equation:**
\[
x^2 + 2x = 8
\]
2. **Complete the square:**
- Take half of the coefficient of \(x\), which is \(2\). Half of \(2\) is \(1\), and squaring it gives \(1^2 = 1\).
- Add and subtract this square (\(1\)) inside the equation.
\[
x^2 + 2x + 1 - 1 = 8
\]
This simplifies to:
\[
(x + 1)^2 - 1 = 8
\]
Now add \(1\) to both sides:
\[
(x + 1)^2 = 9
\]
3. **Take the square root of both sides:**
\[
x + 1 = \pm 3
\]
4. **Solve for \(x\):**
- First solution:
\[
x + 1 = 3 \implies x = 3 - 1 = 2
\]
- Second solution:
\[
x + 1 = -3 \implies x = -3 - 1 = -4
\]
Thus, the solutions are:
\[
x = 2 \quad \text{and} \quad x = -4
\]
Rounding off to the options given, the correct response is:
\[
\boxed{x = 2, \; x = -4}
\]
Answered by
GPT-4o mini
To solve the quadratic equation \(x^2 - 22x + 1 = 15\) by completing the square, we will follow these steps:
1. **Rearrange the equation:**
Move \(15\) to the left side:
\[
x^2 - 22x + 1 - 15 = 0
\]
Simplifying gives:
\[
x^2 - 22x - 14 = 0
\]
2. **Move the constant to the other side:**
\[
x^2 - 22x = 14
\]
3. **Complete the square:**
- Take half of the coefficient of \(x\) (which is \(-22\)), so half of \(-22\) is \(-11\), and squaring it gives \((-11)^2 = 121\).
- Add and subtract this square:
\[
x^2 - 22x + 121 - 121 = 14
\]
This can be rewritten as:
\[
(x - 11)^2 - 121 = 14
\]
Now add \(121\) to both sides:
\[
(x - 11)^2 = 14 + 121 = 135
\]
4. **Take the square root of both sides:**
\[
x - 11 = \pm \sqrt{135}
\]
5. **Solve for \(x\):**
\[
x = 11 \pm \sqrt{135}
\]
This gives us two solutions:
\[
x = 11 + \sqrt{135} \quad \text{and} \quad x = 11 - \sqrt{135}
\]
### Final Answer:
Now, reducing \(\sqrt{135}\):
\[
\sqrt{135} = \sqrt{9 \cdot 15} = 3\sqrt{15}
\]
So the solutions can also be written as:
\[
x = 11 + 3\sqrt{15} \quad \text{and} \quad x = 11 - 3\sqrt{15}
\]
The correct response is:
\[
\boxed{x = 11 + 3\sqrt{15}, \; 11 - 3\sqrt{15}}
\]
1. **Rearrange the equation:**
Move \(15\) to the left side:
\[
x^2 - 22x + 1 - 15 = 0
\]
Simplifying gives:
\[
x^2 - 22x - 14 = 0
\]
2. **Move the constant to the other side:**
\[
x^2 - 22x = 14
\]
3. **Complete the square:**
- Take half of the coefficient of \(x\) (which is \(-22\)), so half of \(-22\) is \(-11\), and squaring it gives \((-11)^2 = 121\).
- Add and subtract this square:
\[
x^2 - 22x + 121 - 121 = 14
\]
This can be rewritten as:
\[
(x - 11)^2 - 121 = 14
\]
Now add \(121\) to both sides:
\[
(x - 11)^2 = 14 + 121 = 135
\]
4. **Take the square root of both sides:**
\[
x - 11 = \pm \sqrt{135}
\]
5. **Solve for \(x\):**
\[
x = 11 \pm \sqrt{135}
\]
This gives us two solutions:
\[
x = 11 + \sqrt{135} \quad \text{and} \quad x = 11 - \sqrt{135}
\]
### Final Answer:
Now, reducing \(\sqrt{135}\):
\[
\sqrt{135} = \sqrt{9 \cdot 15} = 3\sqrt{15}
\]
So the solutions can also be written as:
\[
x = 11 + 3\sqrt{15} \quad \text{and} \quad x = 11 - 3\sqrt{15}
\]
The correct response is:
\[
\boxed{x = 11 + 3\sqrt{15}, \; 11 - 3\sqrt{15}}
\]
Answered by
GPT-4o mini
To solve the quadratic equation \(x^2 - 6x - 16 = 0\) using the quadratic formula, we start by identifying the coefficients \(a\), \(b\), and \(c\):
- \(a = 1\)
- \(b = -6\)
- \(c = -16\)
The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
### Step 1: Calculate the Discriminant
We first calculate the discriminant \(D\):
\[
D = b^2 - 4ac
\]
Substituting the values of \(a\), \(b\), and \(c\):
\[
D = (-6)^2 - 4 \cdot 1 \cdot (-16)
\]
\[
D = 36 + 64
\]
\[
D = 100
\]
### Step 2: Apply the Quadratic Formula
Now substitute the values of \(b\), \(D\), and \(a\) into the quadratic formula:
\[
x = \frac{-(-6) \pm \sqrt{100}}{2 \cdot 1}
\]
\[
x = \frac{6 \pm 10}{2}
\]
### Step 3: Solve for \(x\)
This gives us two potential solutions:
1. For the positive case:
\[
x_1 = \frac{6 + 10}{2} = \frac{16}{2} = 8
\]
2. For the negative case:
\[
x_2 = \frac{6 - 10}{2} = \frac{-4}{2} = -2
\]
### Final Solutions
Thus, the two solutions to the equation \(x^2 - 6x - 16 = 0\) are:
\[
x = 8 \quad \text{and} \quad x = -2
\]
### Correct Response
The correct response is:
\[
\boxed{x = 8, \; x = -2}
\]
- \(a = 1\)
- \(b = -6\)
- \(c = -16\)
The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
### Step 1: Calculate the Discriminant
We first calculate the discriminant \(D\):
\[
D = b^2 - 4ac
\]
Substituting the values of \(a\), \(b\), and \(c\):
\[
D = (-6)^2 - 4 \cdot 1 \cdot (-16)
\]
\[
D = 36 + 64
\]
\[
D = 100
\]
### Step 2: Apply the Quadratic Formula
Now substitute the values of \(b\), \(D\), and \(a\) into the quadratic formula:
\[
x = \frac{-(-6) \pm \sqrt{100}}{2 \cdot 1}
\]
\[
x = \frac{6 \pm 10}{2}
\]
### Step 3: Solve for \(x\)
This gives us two potential solutions:
1. For the positive case:
\[
x_1 = \frac{6 + 10}{2} = \frac{16}{2} = 8
\]
2. For the negative case:
\[
x_2 = \frac{6 - 10}{2} = \frac{-4}{2} = -2
\]
### Final Solutions
Thus, the two solutions to the equation \(x^2 - 6x - 16 = 0\) are:
\[
x = 8 \quad \text{and} \quad x = -2
\]
### Correct Response
The correct response is:
\[
\boxed{x = 8, \; x = -2}
\]
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