Question
Factor the trinomial:
2, x, squared, plus, 5, x, plus, 2
2x
2
+5x+2
2, x, squared, plus, 5, x, plus, 2
2x
2
+5x+2
Answers
Answer
Factor completely.
81, x, squared, minus, 49
81x
2
−49
81, x, squared, minus, 49
81x
2
−49
Answer
Solve for the roots in simplest form using the quadratic formula:
2, x, squared, plus, 54, equals, minus, 24, x
2x
2
+54=
−24x
2, x, squared, plus, 54, equals, minus, 24, x
2x
2
+54=
−24x
Answer
Graph the equation y, equals, minus, x, squared, minus, 4, x, plus, 5y=−x
2
−4x+5 on the accompanying set of axes. You must plot 5 points including the roots and the vertex. Using the graph, determine the vertex of the parabola.
Click to plot points. Click points to delete them.
2
−4x+5 on the accompanying set of axes. You must plot 5 points including the roots and the vertex. Using the graph, determine the vertex of the parabola.
Click to plot points. Click points to delete them.
Answered by
GPT-4o mini
To factor the trinomial \( 2x^2 + 5x + 2 \), we can look for two numbers that multiply to \( 2 \times 2 = 4 \) (the product of the coefficient of \( x^2 \) and the constant term) and add up to \( 5 \) (the coefficient of \( x \)).
The two numbers that satisfy these conditions are \( 4 \) and \( 1 \).
Using these numbers, we can rewrite the middle term \( 5x \) as \( 4x + 1x \):
\[
2x^2 + 4x + 1x + 2
\]
Next, we can group the terms:
\[
(2x^2 + 4x) + (1x + 2)
\]
Now, factor out the common factors in each group:
\[
2x(x + 2) + 1(x + 2)
\]
Notice that \( (x + 2) \) is a common factor:
\[
(2x + 1)(x + 2)
\]
Thus, the factored form of the trinomial \( 2x^2 + 5x + 2 \) is:
\[
(2x + 1)(x + 2)
\]
The two numbers that satisfy these conditions are \( 4 \) and \( 1 \).
Using these numbers, we can rewrite the middle term \( 5x \) as \( 4x + 1x \):
\[
2x^2 + 4x + 1x + 2
\]
Next, we can group the terms:
\[
(2x^2 + 4x) + (1x + 2)
\]
Now, factor out the common factors in each group:
\[
2x(x + 2) + 1(x + 2)
\]
Notice that \( (x + 2) \) is a common factor:
\[
(2x + 1)(x + 2)
\]
Thus, the factored form of the trinomial \( 2x^2 + 5x + 2 \) is:
\[
(2x + 1)(x + 2)
\]
Answered by
GPT-4o mini
The expression \(81x^2 - 49\) is a difference of squares, which can be factored using the formula:
\[
a^2 - b^2 = (a - b)(a + b)
\]
In this case:
- \(a^2 = 81x^2 \implies a = 9x\)
- \(b^2 = 49 \implies b = 7\)
Now, we can apply the difference of squares formula:
\[
81x^2 - 49 = (9x - 7)(9x + 7)
\]
Thus, the completely factored form of \(81x^2 - 49\) is:
\[
(9x - 7)(9x + 7)
\]
\[
a^2 - b^2 = (a - b)(a + b)
\]
In this case:
- \(a^2 = 81x^2 \implies a = 9x\)
- \(b^2 = 49 \implies b = 7\)
Now, we can apply the difference of squares formula:
\[
81x^2 - 49 = (9x - 7)(9x + 7)
\]
Thus, the completely factored form of \(81x^2 - 49\) is:
\[
(9x - 7)(9x + 7)
\]
Answered by
GPT-4o mini
To solve the equation \(2x^2 + 54 = -24x\), we first rearrange it into standard quadratic form \(ax^2 + bx + c = 0\).
Starting with:
\[
2x^2 + 54 + 24x = 0
\]
We can rearrange it to:
\[
2x^2 + 24x + 54 = 0
\]
Next, divide everything by 2 to simplify:
\[
x^2 + 12x + 27 = 0
\]
Now we can apply the quadratic formula, which is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For our equation:
- \(a = 1\)
- \(b = 12\)
- \(c = 27\)
Now we find the discriminant:
\[
b^2 - 4ac = 12^2 - 4 \cdot 1 \cdot 27 = 144 - 108 = 36
\]
Since the discriminant is positive, we have two real roots. Now substituting into the quadratic formula:
\[
x = \frac{-12 \pm \sqrt{36}}{2 \cdot 1}
\]
\[
x = \frac{-12 \pm 6}{2}
\]
This gives us two possible solutions:
1. \(x = \frac{-12 + 6}{2} = \frac{-6}{2} = -3\)
2. \(x = \frac{-12 - 6}{2} = \frac{-18}{2} = -9\)
Thus, the roots of the equation \(2x^2 + 54 = -24x\) are:
\[
\boxed{-3} \quad \text{and} \quad \boxed{-9}
\]
Starting with:
\[
2x^2 + 54 + 24x = 0
\]
We can rearrange it to:
\[
2x^2 + 24x + 54 = 0
\]
Next, divide everything by 2 to simplify:
\[
x^2 + 12x + 27 = 0
\]
Now we can apply the quadratic formula, which is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For our equation:
- \(a = 1\)
- \(b = 12\)
- \(c = 27\)
Now we find the discriminant:
\[
b^2 - 4ac = 12^2 - 4 \cdot 1 \cdot 27 = 144 - 108 = 36
\]
Since the discriminant is positive, we have two real roots. Now substituting into the quadratic formula:
\[
x = \frac{-12 \pm \sqrt{36}}{2 \cdot 1}
\]
\[
x = \frac{-12 \pm 6}{2}
\]
This gives us two possible solutions:
1. \(x = \frac{-12 + 6}{2} = \frac{-6}{2} = -3\)
2. \(x = \frac{-12 - 6}{2} = \frac{-18}{2} = -9\)
Thus, the roots of the equation \(2x^2 + 54 = -24x\) are:
\[
\boxed{-3} \quad \text{and} \quad \boxed{-9}
\]
Answered by
GPT-4o mini
To graph the equation \(y = -x^2 - 4x + 5\), let's start by rewriting the equation in standard form and determining important points, including the vertex and the roots.
### Step 1: Find the Vertex
The vertex of a parabola given by the form \(y = ax^2 + bx + c\) can be calculated using the formula:
\[
x = -\frac{b}{2a}
\]
For the equation \(y = -x^2 - 4x + 5\):
- \(a = -1\)
- \(b = -4\)
Now we can find \(x\):
\[
x = -\frac{-4}{2 \cdot -1} = \frac{4}{-2} = -2
\]
Now substitute \(x = -2\) into the original equation to find the y-coordinate of the vertex:
\[
y = -(-2)^2 - 4(-2) + 5
\]
\[
y = -4 + 8 + 5 = 9
\]
So, the vertex of the parabola is at the point \((-2, 9)\).
### Step 2: Find the Roots
To find the roots, set \(y = 0\):
\[
0 = -x^2 - 4x + 5
\]
Rearranging gives:
\[
x^2 + 4x - 5 = 0
\]
Now we can factor this equation. We look for two numbers that multiply to \(-5\) and add up to \(4\). The factors are \(5\) and \(-1\):
\[
(x + 5)(x - 1) = 0
\]
Thus, the roots are:
\[
x + 5 = 0 \quad \Rightarrow \quad x = -5
\]
\[
x - 1 = 0 \quad \Rightarrow \quad x = 1
\]
### Step 3: Calculate Additional Points
Now, we can calculate a few additional points for a smoother graph. We can evaluate \(y\) at values around the vertex and the roots. Let's calculate:
1. \(x = -3\):
\[
y = -(-3)^2 - 4(-3) + 5 = -9 + 12 + 5 = 8
\]
2. \(x = -1\):
\[
y = -(-1)^2 - 4(-1) + 5 = -1 + 4 + 5 = 8
\]
### Summary of Points
The points to plot on the graph are:
1. Vertex: \((-2, 9)\)
2. Root 1: \((-5, 0)\)
3. Root 2: \( (1, 0)\)
4. Point: \((-3, 8)\)
5. Point: \((-1, 8)\)
### Step 4: Plotting the Points
Now, let's plot the following points:
1. \((-5, 0)\)
2. \((-3, 8)\)
3. \((-2, 9)\)
4. \((-1, 8)\)
5. \((1, 0)\)
### Step 5: Determine the Vertex
The vertex of the parabola is at the point:
\[
\boxed{(-2, 9)}
\]
To create the graph, you would plot these points on the coordinate axes, connecting them to illustrate the parabolic shape characteristic of a quadratic equation. The vertex should be the highest point on the graph since the parabola opens downwards.
### Step 1: Find the Vertex
The vertex of a parabola given by the form \(y = ax^2 + bx + c\) can be calculated using the formula:
\[
x = -\frac{b}{2a}
\]
For the equation \(y = -x^2 - 4x + 5\):
- \(a = -1\)
- \(b = -4\)
Now we can find \(x\):
\[
x = -\frac{-4}{2 \cdot -1} = \frac{4}{-2} = -2
\]
Now substitute \(x = -2\) into the original equation to find the y-coordinate of the vertex:
\[
y = -(-2)^2 - 4(-2) + 5
\]
\[
y = -4 + 8 + 5 = 9
\]
So, the vertex of the parabola is at the point \((-2, 9)\).
### Step 2: Find the Roots
To find the roots, set \(y = 0\):
\[
0 = -x^2 - 4x + 5
\]
Rearranging gives:
\[
x^2 + 4x - 5 = 0
\]
Now we can factor this equation. We look for two numbers that multiply to \(-5\) and add up to \(4\). The factors are \(5\) and \(-1\):
\[
(x + 5)(x - 1) = 0
\]
Thus, the roots are:
\[
x + 5 = 0 \quad \Rightarrow \quad x = -5
\]
\[
x - 1 = 0 \quad \Rightarrow \quad x = 1
\]
### Step 3: Calculate Additional Points
Now, we can calculate a few additional points for a smoother graph. We can evaluate \(y\) at values around the vertex and the roots. Let's calculate:
1. \(x = -3\):
\[
y = -(-3)^2 - 4(-3) + 5 = -9 + 12 + 5 = 8
\]
2. \(x = -1\):
\[
y = -(-1)^2 - 4(-1) + 5 = -1 + 4 + 5 = 8
\]
### Summary of Points
The points to plot on the graph are:
1. Vertex: \((-2, 9)\)
2. Root 1: \((-5, 0)\)
3. Root 2: \( (1, 0)\)
4. Point: \((-3, 8)\)
5. Point: \((-1, 8)\)
### Step 4: Plotting the Points
Now, let's plot the following points:
1. \((-5, 0)\)
2. \((-3, 8)\)
3. \((-2, 9)\)
4. \((-1, 8)\)
5. \((1, 0)\)
### Step 5: Determine the Vertex
The vertex of the parabola is at the point:
\[
\boxed{(-2, 9)}
\]
To create the graph, you would plot these points on the coordinate axes, connecting them to illustrate the parabolic shape characteristic of a quadratic equation. The vertex should be the highest point on the graph since the parabola opens downwards.
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