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Complete the square to identify the two values that solve the following quadratic equation: x^2−16x−13=4 .(1 point) x=Asked by adaleta
Complete the square to identify the two values that solve the following quadratic equation: x2+8x=9
.(1 point)
x=
.(1 point)
x=
Answers
Answered by
adaleta
Using the quadratic formula, find the solution to x2+6x+9=0
.(1 point)
.(1 point)
Answered by
adaleta
Solve −2x2+3x−1=−2
using the quadratic formula. Round the solution(s) to the nearest hundredths. Enter the smaller solution first.(1 point)
x=
or
using the quadratic formula. Round the solution(s) to the nearest hundredths. Enter the smaller solution first.(1 point)
x=
or
Answered by
adaleta
How many real solutions does the following quadratic equation have?
25x2+60x+36=0
(1 point)
solution(s)
25x2+60x+36=0
(1 point)
solution(s)
Answered by
GPT-4o mini
To complete the square for the quadratic equation \( x^2 + 8x = 9 \), follow these steps:
1. **Rearrange the equation** to set it to zero:
\[
x^2 + 8x - 9 = 0
\]
2. **Move the constant to the other side:**
\[
x^2 + 8x = 9
\]
3. **Complete the square**:
- Take half of the coefficient of \( x \) (which is 8), square it, and add it to both sides. Half of 8 is 4, and squaring it gives \( 4^2 = 16 \).
Add 16 to both sides:
\[
x^2 + 8x + 16 = 9 + 16
\]
\[
x^2 + 8x + 16 = 25
\]
4. **Rewrite the left side as a squared term**:
\[
(x + 4)^2 = 25
\]
5. **Take the square root of both sides**:
\[
x + 4 = \pm 5
\]
6. **Solve for \( x \)**:
- For the positive case:
\[
x + 4 = 5 \quad \Rightarrow \quad x = 5 - 4 \quad \Rightarrow \quad x = 1
\]
- For the negative case:
\[
x + 4 = -5 \quad \Rightarrow \quad x = -5 - 4 \quad \Rightarrow \quad x = -9
\]
Thus, the two values that solve the quadratic equation \( x^2 + 8x = 9 \) are:
\[
x = 1 \quad \text{and} \quad x = -9
\]
1. **Rearrange the equation** to set it to zero:
\[
x^2 + 8x - 9 = 0
\]
2. **Move the constant to the other side:**
\[
x^2 + 8x = 9
\]
3. **Complete the square**:
- Take half of the coefficient of \( x \) (which is 8), square it, and add it to both sides. Half of 8 is 4, and squaring it gives \( 4^2 = 16 \).
Add 16 to both sides:
\[
x^2 + 8x + 16 = 9 + 16
\]
\[
x^2 + 8x + 16 = 25
\]
4. **Rewrite the left side as a squared term**:
\[
(x + 4)^2 = 25
\]
5. **Take the square root of both sides**:
\[
x + 4 = \pm 5
\]
6. **Solve for \( x \)**:
- For the positive case:
\[
x + 4 = 5 \quad \Rightarrow \quad x = 5 - 4 \quad \Rightarrow \quad x = 1
\]
- For the negative case:
\[
x + 4 = -5 \quad \Rightarrow \quad x = -5 - 4 \quad \Rightarrow \quad x = -9
\]
Thus, the two values that solve the quadratic equation \( x^2 + 8x = 9 \) are:
\[
x = 1 \quad \text{and} \quad x = -9
\]
Answered by
GPT-4o mini
To solve the quadratic equation \( x^2 + 6x + 9 = 0 \) using the quadratic formula, we start with the general form of a quadratic equation:
\[
ax^2 + bx + c = 0
\]
For the given equation:
- \( a = 1 \)
- \( b = 6 \)
- \( c = 9 \)
The quadratic formula is given by:
\[
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}
\]
Now, we substitute the values of \( a \), \( b \), and \( c \) into the formula:
1. **Calculate the discriminant** (\( b^2 - 4ac \)):
\[
b^2 - 4ac = 6^2 - 4(1)(9) = 36 - 36 = 0
\]
2. **Plug values into the quadratic formula**:
\[
x = \frac{{-6 \pm \sqrt{0}}}{{2(1)}}
\]
\[
x = \frac{{-6 \pm 0}}{2} = \frac{{-6}}{2} = -3
\]
Since the discriminant is 0, there is only one real solution.
Thus, the solution to the equation \( x^2 + 6x + 9 = 0 \) is:
\[
x = -3
\]
\[
ax^2 + bx + c = 0
\]
For the given equation:
- \( a = 1 \)
- \( b = 6 \)
- \( c = 9 \)
The quadratic formula is given by:
\[
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}
\]
Now, we substitute the values of \( a \), \( b \), and \( c \) into the formula:
1. **Calculate the discriminant** (\( b^2 - 4ac \)):
\[
b^2 - 4ac = 6^2 - 4(1)(9) = 36 - 36 = 0
\]
2. **Plug values into the quadratic formula**:
\[
x = \frac{{-6 \pm \sqrt{0}}}{{2(1)}}
\]
\[
x = \frac{{-6 \pm 0}}{2} = \frac{{-6}}{2} = -3
\]
Since the discriminant is 0, there is only one real solution.
Thus, the solution to the equation \( x^2 + 6x + 9 = 0 \) is:
\[
x = -3
\]
Answered by
GPT-4o mini
To solve the equation \(-2x^2 + 3x - 1 = -2\) using the quadratic formula, first rearrange the equation to set it to zero:
\[
-2x^2 + 3x - 1 + 2 = 0
\]
This simplifies to:
\[
-2x^2 + 3x + 1 = 0
\]
Now, we can identify the coefficients \(a\), \(b\), and \(c\):
- \(a = -2\)
- \(b = 3\)
- \(c = 1\)
The quadratic formula is given by:
\[
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}
\]
Substituting in the values of \(a\), \(b\), and \(c\):
1. **Calculate the discriminant** \((b^2 - 4ac)\):
\[
b^2 - 4ac = 3^2 - 4(-2)(1) = 9 + 8 = 17
\]
2. **Plug these values into the quadratic formula**:
\[
x = \frac{{-3 \pm \sqrt{17}}}{{2(-2)}}
\]
\[
x = \frac{{-3 \pm \sqrt{17}}}{{-4}}
\]
3. **Evaluate the two potential solutions**:
- For the plus case:
\[
x_1 = \frac{{-3 + \sqrt{17}}}{{-4}}
\]
- For the minus case:
\[
x_2 = \frac{{-3 - \sqrt{17}}}{{-4}}
\]
4. **Calculate the numeric values**:
First, calculate \(\sqrt{17} \approx 4.123\).
Now plug this value into the formulas:
\[
x_1 = \frac{{-3 + 4.123}}{{-4}} \approx \frac{{1.123}}{{-4}} \approx -0.28075 \quad (\text{approximately } -0.28)
\]
\[
x_2 = \frac{{-3 - 4.123}}{{-4}} \approx \frac{{-7.123}}{{-4}} \approx 1.78075 \quad (\text{approximately } 1.78)
\]
Therefore, rounding both solutions to the nearest hundredths, we find:
- Smaller solution: \( x \approx -0.28 \)
- Larger solution: \( x \approx 1.78 \)
Thus, the two solutions are:
\[
x \approx -0.28 \quad \text{or} \quad x \approx 1.78
\]
\[
-2x^2 + 3x - 1 + 2 = 0
\]
This simplifies to:
\[
-2x^2 + 3x + 1 = 0
\]
Now, we can identify the coefficients \(a\), \(b\), and \(c\):
- \(a = -2\)
- \(b = 3\)
- \(c = 1\)
The quadratic formula is given by:
\[
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}
\]
Substituting in the values of \(a\), \(b\), and \(c\):
1. **Calculate the discriminant** \((b^2 - 4ac)\):
\[
b^2 - 4ac = 3^2 - 4(-2)(1) = 9 + 8 = 17
\]
2. **Plug these values into the quadratic formula**:
\[
x = \frac{{-3 \pm \sqrt{17}}}{{2(-2)}}
\]
\[
x = \frac{{-3 \pm \sqrt{17}}}{{-4}}
\]
3. **Evaluate the two potential solutions**:
- For the plus case:
\[
x_1 = \frac{{-3 + \sqrt{17}}}{{-4}}
\]
- For the minus case:
\[
x_2 = \frac{{-3 - \sqrt{17}}}{{-4}}
\]
4. **Calculate the numeric values**:
First, calculate \(\sqrt{17} \approx 4.123\).
Now plug this value into the formulas:
\[
x_1 = \frac{{-3 + 4.123}}{{-4}} \approx \frac{{1.123}}{{-4}} \approx -0.28075 \quad (\text{approximately } -0.28)
\]
\[
x_2 = \frac{{-3 - 4.123}}{{-4}} \approx \frac{{-7.123}}{{-4}} \approx 1.78075 \quad (\text{approximately } 1.78)
\]
Therefore, rounding both solutions to the nearest hundredths, we find:
- Smaller solution: \( x \approx -0.28 \)
- Larger solution: \( x \approx 1.78 \)
Thus, the two solutions are:
\[
x \approx -0.28 \quad \text{or} \quad x \approx 1.78
\]
Answered by
GPT-4o mini
To determine how many real solutions the quadratic equation \( 25x^2 + 60x + 36 = 0 \) has, we can calculate the discriminant \( D \).
The discriminant is given by the formula:
\[
D = b^2 - 4ac
\]
For the given equation, identify the coefficients:
- \( a = 25 \)
- \( b = 60 \)
- \( c = 36 \)
Substituting these values into the discriminant formula:
\[
D = 60^2 - 4(25)(36)
\]
\[
D = 3600 - 3600
\]
\[
D = 0
\]
The discriminant \( D = 0 \) indicates that the quadratic equation has **exactly one real solution** (also called a repeated root).
Thus, the answer is:
- **One real solution**.
The discriminant is given by the formula:
\[
D = b^2 - 4ac
\]
For the given equation, identify the coefficients:
- \( a = 25 \)
- \( b = 60 \)
- \( c = 36 \)
Substituting these values into the discriminant formula:
\[
D = 60^2 - 4(25)(36)
\]
\[
D = 3600 - 3600
\]
\[
D = 0
\]
The discriminant \( D = 0 \) indicates that the quadratic equation has **exactly one real solution** (also called a repeated root).
Thus, the answer is:
- **One real solution**.