Question
The quadratic equation x2+4x−32=0
can be solved by factoring. Which of the following is the factored form?(1 point)
Responses
(x+8)(x−4)=0
left parenthesis x plus 8 right parenthesis left parenthesis x minus 4 right parenthesis equals 0
(x−4)(x−8)=0
left parenthesis x minus 4 right parenthesis left parenthesis x minus 8 right parenthesis equals 0
(x+4)(x+8)=0
left parenthesis x plus 4 right parenthesis left parenthesis x plus 8 right parenthesis equals 0
(x+4)(x−8)=0
can be solved by factoring. Which of the following is the factored form?(1 point)
Responses
(x+8)(x−4)=0
left parenthesis x plus 8 right parenthesis left parenthesis x minus 4 right parenthesis equals 0
(x−4)(x−8)=0
left parenthesis x minus 4 right parenthesis left parenthesis x minus 8 right parenthesis equals 0
(x+4)(x+8)=0
left parenthesis x plus 4 right parenthesis left parenthesis x plus 8 right parenthesis equals 0
(x+4)(x−8)=0
Answers
Answer
Which of the following equations has exactly the same solutions as (x−7)(x+9)=0
?(1 point)
Responses
x2=−2x+63
x squared equals negative 2 x plus 63
x2−2x−63=0
x squared minus 2 x minus 63 equals 0
x2=2x−63
x squared equals 2 x minus 63
x2+2x=−63
?(1 point)
Responses
x2=−2x+63
x squared equals negative 2 x plus 63
x2−2x−63=0
x squared minus 2 x minus 63 equals 0
x2=2x−63
x squared equals 2 x minus 63
x2+2x=−63
Answer
What should be the value of b in x2+bx−55=0
so that its solutions are −11
and 5? (1 point)
Responses
−16
negative 16
16
16
−6
negative 6
6
so that its solutions are −11
and 5? (1 point)
Responses
−16
negative 16
16
16
−6
negative 6
6
Answer
Solve the following quadratic equation using the perfect square trinomial pattern: x2−16x+64=0
.(1 point)
Responses
x=8
x equals 8
x=−64
x equals negative 64
x=64
x equals 64
x=−8
.(1 point)
Responses
x=8
x equals 8
x=−64
x equals negative 64
x=64
x equals 64
x=−8
Answer
Gabriel was asked to solve x2−36=0
by factoring using the difference of squares pattern. What is/are the correct solution(s)?(1 point)
Responses
x=−6
x equals negative 6
x=18
, x=−18
x equals 18 , x equals negative 18
x=6
, x=−6
x equals 6 , x equals negative 6
x=6
by factoring using the difference of squares pattern. What is/are the correct solution(s)?(1 point)
Responses
x=−6
x equals negative 6
x=18
, x=−18
x equals 18 , x equals negative 18
x=6
, x=−6
x equals 6 , x equals negative 6
x=6
Answered by
GPT-4o mini
To factor the quadratic equation \(x^2 + 4x - 32 = 0\), we can look for two numbers that multiply to \(-32\) (the constant term) and add to \(4\) (the coefficient of the linear term).
The numbers \(8\) and \(-4\) fit this requirement because:
- \(8 \times -4 = -32\)
- \(8 + (-4) = 4\)
Thus, the factored form of the given quadratic equation is:
\[
(x + 8)(x - 4) = 0
\]
So, the correct response is:
\[
(x+8)(x−4)=0
\]
The numbers \(8\) and \(-4\) fit this requirement because:
- \(8 \times -4 = -32\)
- \(8 + (-4) = 4\)
Thus, the factored form of the given quadratic equation is:
\[
(x + 8)(x - 4) = 0
\]
So, the correct response is:
\[
(x+8)(x−4)=0
\]
Answered by
GPT-4o mini
To find which of the given equations has exactly the same solutions as \((x-7)(x+9)=0\), we need to identify what solutions this equation has first.
Setting \((x-7)(x+9)=0\) gives us two solutions:
1. \(x - 7 = 0 \Rightarrow x = 7\)
2. \(x + 9 = 0 \Rightarrow x = -9\)
Now we will check each of the provided equations to see which has the same solutions:
1. **\(x^2 = -2x + 63\)**
- Rearranging gives us \(x^2 + 2x - 63 = 0\), which can be factored as \((x - 7)(x + 9) = 0\). This has the same solutions: \(7\) and \(-9\).
2. **\(x^2 - 2x - 63 = 0\)**
- This can be factored as \((x - 9)(x + 7) = 0\), which has solutions \(9\) and \(-7\). This does not have the same solutions.
3. **\(x^2 = 2x - 63\)**
- Rearranging gives us \(x^2 - 2x + 63 = 0\), which does not have the same factorization as before.
4. **\(x^2 + 2x = -63\)**
- Rearranging gives us \(x^2 + 2x + 63 = 0\), which also does not have the same factorization as before.
The only equation that simplifies to the same factors as the original equation is:
\[
x^2 = -2x + 63
\]
Thus, the correct response is:
**\(x^2 = -2x + 63\)**.
Setting \((x-7)(x+9)=0\) gives us two solutions:
1. \(x - 7 = 0 \Rightarrow x = 7\)
2. \(x + 9 = 0 \Rightarrow x = -9\)
Now we will check each of the provided equations to see which has the same solutions:
1. **\(x^2 = -2x + 63\)**
- Rearranging gives us \(x^2 + 2x - 63 = 0\), which can be factored as \((x - 7)(x + 9) = 0\). This has the same solutions: \(7\) and \(-9\).
2. **\(x^2 - 2x - 63 = 0\)**
- This can be factored as \((x - 9)(x + 7) = 0\), which has solutions \(9\) and \(-7\). This does not have the same solutions.
3. **\(x^2 = 2x - 63\)**
- Rearranging gives us \(x^2 - 2x + 63 = 0\), which does not have the same factorization as before.
4. **\(x^2 + 2x = -63\)**
- Rearranging gives us \(x^2 + 2x + 63 = 0\), which also does not have the same factorization as before.
The only equation that simplifies to the same factors as the original equation is:
\[
x^2 = -2x + 63
\]
Thus, the correct response is:
**\(x^2 = -2x + 63\)**.
Answered by
GPT-4o mini
To find the value of \(b\) in the equation \(x^2 + bx - 55 = 0\) such that its solutions are \(-11\) and \(5\), we can use Vieta's formulas, which state that for a quadratic equation of the form \(x^2 + bx + c = 0\), the sum of the solutions \(-\frac{b}{a}\) and the product of the solutions \(\frac{c}{a}\).
For our equation:
1. The product of the solutions (\(-11\) and \(5\)) must equal \(c\) (in this case \(-55\)):
\[
-11 \times 5 = -55
\]
This is correct.
2. The sum of the solutions must equal \(-b\):
\[
-11 + 5 = -6
\]
Therefore:
\[
-b = -6 \rightarrow b = 6
\]
Thus, the value of \(b\) should be \(6\).
The correct response is:
**6**.
For our equation:
1. The product of the solutions (\(-11\) and \(5\)) must equal \(c\) (in this case \(-55\)):
\[
-11 \times 5 = -55
\]
This is correct.
2. The sum of the solutions must equal \(-b\):
\[
-11 + 5 = -6
\]
Therefore:
\[
-b = -6 \rightarrow b = 6
\]
Thus, the value of \(b\) should be \(6\).
The correct response is:
**6**.
Answered by
GPT-4o mini
To solve the quadratic equation \(x^2 - 16x + 64 = 0\) using the perfect square trinomial pattern, we can rewrite the equation.
We notice that \(x^2 - 16x + 64\) can be expressed as:
\[
(x - 8)^2 = 0
\]
This form follows the pattern \((a - b)^2 = a^2 - 2ab + b^2\). Here, \(a = x\) and \(b = 8\), since:
\[
(x - 8)(x - 8) = x^2 - 8x - 8x + 64 = x^2 - 16x + 64
\]
Now we can set the equation to zero:
\[
(x - 8)^2 = 0
\]
To solve for \(x\), take the square root of both sides:
\[
x - 8 = 0
\]
Thus, solving for \(x\) gives us:
\[
x = 8
\]
So the solution to the equation is:
**x = 8**.
We notice that \(x^2 - 16x + 64\) can be expressed as:
\[
(x - 8)^2 = 0
\]
This form follows the pattern \((a - b)^2 = a^2 - 2ab + b^2\). Here, \(a = x\) and \(b = 8\), since:
\[
(x - 8)(x - 8) = x^2 - 8x - 8x + 64 = x^2 - 16x + 64
\]
Now we can set the equation to zero:
\[
(x - 8)^2 = 0
\]
To solve for \(x\), take the square root of both sides:
\[
x - 8 = 0
\]
Thus, solving for \(x\) gives us:
\[
x = 8
\]
So the solution to the equation is:
**x = 8**.
Answered by
GPT-4o mini
To solve the equation \(x^2 - 36 = 0\) by factoring using the difference of squares pattern, we can recognize that the equation can be factored as follows:
\[
x^2 - 36 = (x - 6)(x + 6) = 0
\]
Next, we can set each factor equal to zero to find the solutions:
1. \(x - 6 = 0\) gives:
\[
x = 6
\]
2. \(x + 6 = 0\) gives:
\[
x = -6
\]
Thus, the solutions to the equation \(x^2 - 36 = 0\) are:
\[
x = 6 \quad \text{and} \quad x = -6
\]
Therefore, the correct response is:
**x = 6, x = -6**.
\[
x^2 - 36 = (x - 6)(x + 6) = 0
\]
Next, we can set each factor equal to zero to find the solutions:
1. \(x - 6 = 0\) gives:
\[
x = 6
\]
2. \(x + 6 = 0\) gives:
\[
x = -6
\]
Thus, the solutions to the equation \(x^2 - 36 = 0\) are:
\[
x = 6 \quad \text{and} \quad x = -6
\]
Therefore, the correct response is:
**x = 6, x = -6**.