Responses
x=−4 and x=2
x equals negative 4 and x equals 2
x=−2 and x=−4
x equals negative 2 and x equals negative 4
x=−2 and x=4
x equals negative 2 and x equals 4
x=2 and x=4
11 answers
To solve the equation \((x+4)(x−2)=0\), we need to set each factor equal to zero:
\(x + 4 = 0\)
\(x - 2 = 0\)
Thus, the values of \(x\) that make the equation equal zero are \(x = -4\) and \(x = 2\).
The correct response is: x equals negative 4 and x equals 2.
To solve the factored quadratic equation \((19x−3)(9+x)=0\), we can set each factor to zero:
First factor:
\(19x - 3 = 0\)
Solving for \(x\), we get:
\(19x = 3\)
\(x = \frac{3}{19}\)
Second factor:
\(9 + x = 0\)
Solving for \(x\), we get:
\(x = -9\)
Thus, the solution set is \(x = -9\) and \(x = \frac{3}{19}\).
The correct response is: The solution set is x={−9,3/19}.
To solve the quadratic equation \(84x^2 + 16x = 0\) by factoring out the greatest common factor (GCF), we first identify the GCF of the two terms:
The GCF of \(84x^2\) and \(16x\) is \(4x\).
Now, we factor \(4x\) out of the equation:
\[ 84x^2 + 16x = 4x(21x + 4) = 0 \]
Next, we can set each factor equal to zero:
For the first factor: \[ 4x = 0 \implies x = 0 \]
For the second factor: \[ 21x + 4 = 0 \implies 21x = -4 \implies x = -\frac{4}{21} \]
Thus, the solutions are \(x = 0\) and \(x = -\frac{4}{21}\).
The correct response is: The solutions are x=0 and x=421.
To determine which quadratic equation is not solvable by grouping, we usually look for the ability to factor the quadratic expression into two binomials. Grouping typically works best when the equation can be rearranged into two pairs that share a common factor.
Let’s analyze each equation:
\(2x^2 + 14x + 12 = 0\)
This can be factored as:
\[
2(x^2 + 7x + 6) = 2(x + 6)(x + 1) = 0
\]
So this can be solved by grouping.
\(2x^2 - 2x - 10 = 0\)
This can be simplified by factoring out a GCF first:
\[
2(x^2 - x - 5) = 0
\]
The quadratic \(x^2 - x - 5\) can be factored, but the roots are not rational. However, it can still be solved using the quadratic formula.
\(x^2 - 12x + 35 = 0\)
This can be factored as:
\[
(x - 5)(x - 7) = 0
\]
So this can be solved by grouping.
\(x^2 - 2x + 1 = 0\)
This can be factored as:
\[
(x - 1)(x - 1) = (x - 1)^2 = 0
\]
So this can also be solved by grouping.
While all equations can ultimately be solved, the second equation, \(2x^2 - 2x - 10 = 0\), is less straightforward regarding traditional grouping, primarily because it does not factor nicely into simple integers. The others can be grouped or factored directly into products of linear terms.
Thus, the quadratic equation that is not solvable by grouping in a straightforward manner is: \(2x^2 - 2x - 10 = 0\).
Let's review Jedida's steps for solving the quadratic equation \(x^2 - 96 = 4x\):
\[ x^2 - 96 = 4x \]
She rearranged the equation by moving \(4x\) to the left side: \[ x^2 + 4x - 96 = 0 \] This step is correct.
She factored the quadratic: \[ (x + 12)(x - 8) = 0 \] To factor correctly, we can verify that: \[ x^2 + 4x - 96 = (x + 12)(x - 8) \] Expanding \( (x + 12)(x - 8) \): \[ x^2 - 8x + 12x - 96 = x^2 + 4x - 96 \] So, this factorization is also correct.
Using the Zero Product Property: \[ x + 12 = 0 \quad \text{or} \quad x - 8 = 0 \] This leads to: \[ x = -12 \quad \text{or} \quad x = 8 \] This part is also correct.
Jedida did not make any mistakes in her steps. Therefore, the correct response is: She did not make any mistakes.
To solve the quadratic equation \(x^2 + x + 0.25 = 0\) using the perfect square trinomial pattern, we first recognize that the equation can be rewritten in the form \((x + a)^2 = b\).
Identify the perfect square trinomial: The expression \(x^2 + x + 0.25\) can be written as: \[ x^2 + 2 \cdot \frac{1}{2} \cdot x + \left(\frac{1}{2}\right)^2 \] Here, \(a = \frac{1}{2}\).
Rewrite the equation as a perfect square: This allows us to rewrite it as: \[ \left(x + \frac{1}{2}\right)^2 = 0 \]
Solve for \(x\): Now, taking the square root of both sides: \[ x + \frac{1}{2} = 0 \] Thus, \[ x = -\frac{1}{2} \]
So, the solution to the equation \(x^2 + x + 0.25 = 0\) is: \[ x = -0.5 \]
The correct response is: x equals negative 0.5.
