To simplify, we use the distributive property and multiply each term of the first factor (x + 5) by each term of the second factor (2x^2 + 2x - 3):
x(2x^2) + x(2x) + x(-3) + 5(2x^2) + 5(2x) + 5(-3)
Simplifying each term:
2x^3 + 2x^2 - 3x + 10x^2 + 10x - 15
Combining like terms:
2x^3 + 12x^2 + 7x - 15
Therefore, the polynomial in standard form is:
2x^3 + 12x^2 + 7x - 15.
simplify the expression to a polynomial in standard form: (x + 5)(2x^2 + 2x - 3)
25 answers
find the coordinates of the vert of the following parabola using the graph technology. write your answer as an (x, y) point. y = x^2 + 6x + 18
To find the vertex of the parabola using graph technology, we can either complete the square or use the vertex formula.
Completing the square:
y = x^2 + 6x + 18
y + 9 = x^2 + 6x + 9 + 18
y + 9 = (x + 3)^2 + 18
y = (x + 3)^2 + 9
Now, we can see that the vertex of the parabola is (-3, 9).
Using the vertex formula:
Recall that the vertex of the parabola y = ax^2 + bx + c is given by (-b/2a, f(-b/2a)), where f(-b/2a) is the y-coordinate of the vertex.
In this case, a = 1, b = 6, and c = 18.
The x-coordinate of the vertex is -b/2a = -6/2 = -3.
To find the y-coordinate, we substitute x = -3 into the equation:
y = (-3)^2 + 6(-3) + 18 = 9
So the vertex is (-3, 9).
Completing the square:
y = x^2 + 6x + 18
y + 9 = x^2 + 6x + 9 + 18
y + 9 = (x + 3)^2 + 18
y = (x + 3)^2 + 9
Now, we can see that the vertex of the parabola is (-3, 9).
Using the vertex formula:
Recall that the vertex of the parabola y = ax^2 + bx + c is given by (-b/2a, f(-b/2a)), where f(-b/2a) is the y-coordinate of the vertex.
In this case, a = 1, b = 6, and c = 18.
The x-coordinate of the vertex is -b/2a = -6/2 = -3.
To find the y-coordinate, we substitute x = -3 into the equation:
y = (-3)^2 + 6(-3) + 18 = 9
So the vertex is (-3, 9).
solve for all values of x by factoring. x^2 - 2x - 45 = 2x
First, we simplify the equation by moving all terms to one side:
x^2 - 2x - 45 - 2x = 0
Simplifying:
x^2 - 4x - 45 = 0
To solve for x by factoring, we need to find two numbers that multiply to -45 and add to -4. These numbers are -9 and 5, since (-9)(5) = -45 and (-9) + 5 = -4.
So, we can rewrite the quadratic expression as:
x^2 - 9x + 5x - 45 = 0
Now, we can factor by grouping:
(x^2 - 9x) + (5x - 45) = 0
x(x - 9) + 5(x - 9) = 0
(x + 5)(x - 9) = 0
This gives us two solutions: x = -5 or x = 9.
Therefore, the solution set is {-5, 9}.
x^2 - 2x - 45 - 2x = 0
Simplifying:
x^2 - 4x - 45 = 0
To solve for x by factoring, we need to find two numbers that multiply to -45 and add to -4. These numbers are -9 and 5, since (-9)(5) = -45 and (-9) + 5 = -4.
So, we can rewrite the quadratic expression as:
x^2 - 9x + 5x - 45 = 0
Now, we can factor by grouping:
(x^2 - 9x) + (5x - 45) = 0
x(x - 9) + 5(x - 9) = 0
(x + 5)(x - 9) = 0
This gives us two solutions: x = -5 or x = 9.
Therefore, the solution set is {-5, 9}.
sole the following quadratic equation for all values of x in the simplest form. 4(x + 4)^2 - 36 = 28
Starting with the given equation:
4(x + 4)^2 - 36 = 28
We first simplify the left-hand side by distributing the 4:
4(x^2 + 8x + 16) - 36 = 28
Simplifying:
4x^2 + 32x + 4 - 36 = 28
4x^2 + 32x - 60 = 0
We can simplify further by dividing both sides by 4:
x^2 + 8x - 15 = 0
Now, we can factor by finding two numbers that multiply to -15 and add to 8. These numbers are 5 and -3, since 5(-3) = -15 and 5 + (-3) = 8.
So we can rewrite the quadratic expression as:
(x + 5)(x - 3) = 0
This gives us two solutions:
x + 5 = 0 or x - 3 = 0
Solving for x:
x = -5 or x = 3
Therefore, the solution set is {-5, 3}.
4(x + 4)^2 - 36 = 28
We first simplify the left-hand side by distributing the 4:
4(x^2 + 8x + 16) - 36 = 28
Simplifying:
4x^2 + 32x + 4 - 36 = 28
4x^2 + 32x - 60 = 0
We can simplify further by dividing both sides by 4:
x^2 + 8x - 15 = 0
Now, we can factor by finding two numbers that multiply to -15 and add to 8. These numbers are 5 and -3, since 5(-3) = -15 and 5 + (-3) = 8.
So we can rewrite the quadratic expression as:
(x + 5)(x - 3) = 0
This gives us two solutions:
x + 5 = 0 or x - 3 = 0
Solving for x:
x = -5 or x = 3
Therefore, the solution set is {-5, 3}.
sole the following quadratic equation for all values of x in the simplest form. 5(x - 6)^2 - 29 = -19
Starting with the given equation:
5(x - 6)^2 - 29 = -19
We first simplify by adding 29 to both sides:
5(x - 6)^2 = 10
Now we can divide both sides by 5:
(x - 6)^2 = 2
Taking the square root of both sides, we get:
x - 6 = ±√2
Solving for x:
x = 6 ± √2
Therefore, the solution set is {6 + √2, 6 - √2}.
5(x - 6)^2 - 29 = -19
We first simplify by adding 29 to both sides:
5(x - 6)^2 = 10
Now we can divide both sides by 5:
(x - 6)^2 = 2
Taking the square root of both sides, we get:
x - 6 = ±√2
Solving for x:
x = 6 ± √2
Therefore, the solution set is {6 + √2, 6 - √2}.
solve the equation for all the values of x by completing the square. x^2 - 8x + 7 = 0
To solve this equation by completing the square, we want to rewrite the left-hand side as a perfect square trinomial.
First, we move the constant term to the right-hand side:
x^2 - 8x = -7
Next, we need to add and subtract a constant term to complete the square. To determine which constant term to use, recall that the middle coefficient is -8. We want to find a constant c such that (c/2)^2 = c^2/4 = 16. Then we can add c^2/4 to both sides to complete the square.
To find c, we can take half of the middle coefficient and square it:
(-8/2)^2 = 16
So, we add and subtract 16:
x^2 - 8x + 16 - 16 = -7
The left-hand side is now a perfect square trinomial:
(x - 4)^2 - 23 = 0
Adding 23 to both sides:
(x - 4)^2 = 23
Taking the square root of both sides:
x - 4 = ±√23
Solving for x:
x = 4 ± √23
Therefore, the solution set is {4 + √23, 4 - √23}.
First, we move the constant term to the right-hand side:
x^2 - 8x = -7
Next, we need to add and subtract a constant term to complete the square. To determine which constant term to use, recall that the middle coefficient is -8. We want to find a constant c such that (c/2)^2 = c^2/4 = 16. Then we can add c^2/4 to both sides to complete the square.
To find c, we can take half of the middle coefficient and square it:
(-8/2)^2 = 16
So, we add and subtract 16:
x^2 - 8x + 16 - 16 = -7
The left-hand side is now a perfect square trinomial:
(x - 4)^2 - 23 = 0
Adding 23 to both sides:
(x - 4)^2 = 23
Taking the square root of both sides:
x - 4 = ±√23
Solving for x:
x = 4 ± √23
Therefore, the solution set is {4 + √23, 4 - √23}.
what is the discrimination of the quadratic equation?
a. -223
b. -225
c. 223
d. 225
a. -223
b. -225
c. 223
d. 225
The discrimination of the quadratic equation is given by the expression b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0.
For example, in the quadratic equation x^2 + 3x + 2 = 0, the coefficients are a = 1, b = 3, and c = 2, so the discrimination is given by:
b^2 - 4ac = 3^2 - 4(1)(2) = 1
Now, considering the answer choices, we can see that the discrimination must be positive because the discriminant of a quadratic equation determines the nature of the roots. Specifically, if the discrimination is greater than zero, then there are two distinct real roots; if it is zero, then there is one real root (with multiplicity 2); and if it is less than zero, there are two complex roots.
Therefore, we can eliminate options (a) and (c) which are negative.
Next, we can compare options (b) and (d), which are both positive.
Using the quadratic equation ax^2 + bx + c = 0,
a=1,
b=-2,
c=-15
Now, the discrimination is:
b^2 - 4ac = (-2)^2 - 4(1)(-15) = 64
Option (d), 225, is the square of 15, which is plausible since 15^2 is positive. We can also check our work by calculating the roots of the quadratic equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac))/2a
Substituting the coefficients a = 1, b = -2, and c = -15, we get:
x = (2 ± √64)/2
Thus, the roots are x = 1 + √(64)/2 and x = 1 - √(64)/2, which simplify to x = 8/2 = 4 and x = -6/2 = -3.
Since the discrimination is positive and equal to 225, the correct answer is (d).
For example, in the quadratic equation x^2 + 3x + 2 = 0, the coefficients are a = 1, b = 3, and c = 2, so the discrimination is given by:
b^2 - 4ac = 3^2 - 4(1)(2) = 1
Now, considering the answer choices, we can see that the discrimination must be positive because the discriminant of a quadratic equation determines the nature of the roots. Specifically, if the discrimination is greater than zero, then there are two distinct real roots; if it is zero, then there is one real root (with multiplicity 2); and if it is less than zero, there are two complex roots.
Therefore, we can eliminate options (a) and (c) which are negative.
Next, we can compare options (b) and (d), which are both positive.
Using the quadratic equation ax^2 + bx + c = 0,
a=1,
b=-2,
c=-15
Now, the discrimination is:
b^2 - 4ac = (-2)^2 - 4(1)(-15) = 64
Option (d), 225, is the square of 15, which is plausible since 15^2 is positive. We can also check our work by calculating the roots of the quadratic equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac))/2a
Substituting the coefficients a = 1, b = -2, and c = -15, we get:
x = (2 ± √64)/2
Thus, the roots are x = 1 + √(64)/2 and x = 1 - √(64)/2, which simplify to x = 8/2 = 4 and x = -6/2 = -3.
Since the discrimination is positive and equal to 225, the correct answer is (d).
graph the equation y = x^2 - 14x + 48 on the accompanying sex of axes. you must plot 5 points including the roots and the vertex
To graph the equation y = x^2 - 14x + 48, we can start by finding the x-intercepts and the vertex.
x-intercepts:
y = 0:
x^2 - 14x + 48 = 0
We can factor this quadratic equation as follows:
(x - 6)(x - 8) = 0
So the x-intercepts are x = 6 and x = 8.
Vertex:
Recall that the x-coordinate of the vertex is given by -b/2a, where a is the coefficient of x^2 and b is the coefficient of x. In this case, a = 1 and b = -14, so the x-coordinate of the vertex is x = -(-14)/(2*1) = 7.
To find the y-coordinate, we can substitute x = 7 into the equation:
y = 7^2 - 14(7) + 48 = -7
So the vertex is (7, -7).
Now, we can plot these points and draw the parabola:
- x = 6, y = 0
- x = 8, y = 0
- x = 7, y = -7
- x = 3, y = 3
- x = 11, y = 3
Note that we can find these additional points by choosing x-values on either side of the vertex and finding the corresponding y-values by substituting into the equation.
The graph should look like this:
x-intercepts:
y = 0:
x^2 - 14x + 48 = 0
We can factor this quadratic equation as follows:
(x - 6)(x - 8) = 0
So the x-intercepts are x = 6 and x = 8.
Vertex:
Recall that the x-coordinate of the vertex is given by -b/2a, where a is the coefficient of x^2 and b is the coefficient of x. In this case, a = 1 and b = -14, so the x-coordinate of the vertex is x = -(-14)/(2*1) = 7.
To find the y-coordinate, we can substitute x = 7 into the equation:
y = 7^2 - 14(7) + 48 = -7
So the vertex is (7, -7).
Now, we can plot these points and draw the parabola:
- x = 6, y = 0
- x = 8, y = 0
- x = 7, y = -7
- x = 3, y = 3
- x = 11, y = 3
Note that we can find these additional points by choosing x-values on either side of the vertex and finding the corresponding y-values by substituting into the equation.
The graph should look like this:
Graph the equation y = x^2 - 4x - 3 on the accompanying sex of axes. You must plot 5 points including the roots and the vertex.
To graph the equation y = x^2 - 4x - 3, we can start by finding the x-intercepts and the vertex.
x-intercepts:
y = 0:
x^2 - 4x - 3 = 0
We can factor this quadratic equation as follows:
(x - 3)(x + 1) = 0
So the x-intercepts are x = 3 and x = -1.
Vertex:
Recall that the x-coordinate of the vertex is given by -b/2a, where a is the coefficient of x^2 and b is the coefficient of x. In this case, a = 1 and b = -4, so the x-coordinate of the vertex is x = -(-4)/(2*1) = 2.
To find the y-coordinate, we can substitute x = 2 into the equation:
y = 2^2 - 4(2) - 3 = -3
So the vertex is (2, -3).
Now, we can plot these points and draw the parabola:
- x = 3, y = 0
- x = -1, y = 0
- x = 2, y = -3
- x = 0, y = -3
- x = 4, y = 5
Note that we can find these additional points by choosing x-values on either side of the vertex and finding the corresponding y-values by substituting into the equation.
The graph should look like this:
x-intercepts:
y = 0:
x^2 - 4x - 3 = 0
We can factor this quadratic equation as follows:
(x - 3)(x + 1) = 0
So the x-intercepts are x = 3 and x = -1.
Vertex:
Recall that the x-coordinate of the vertex is given by -b/2a, where a is the coefficient of x^2 and b is the coefficient of x. In this case, a = 1 and b = -4, so the x-coordinate of the vertex is x = -(-4)/(2*1) = 2.
To find the y-coordinate, we can substitute x = 2 into the equation:
y = 2^2 - 4(2) - 3 = -3
So the vertex is (2, -3).
Now, we can plot these points and draw the parabola:
- x = 3, y = 0
- x = -1, y = 0
- x = 2, y = -3
- x = 0, y = -3
- x = 4, y = 5
Note that we can find these additional points by choosing x-values on either side of the vertex and finding the corresponding y-values by substituting into the equation.
The graph should look like this:
solve for all values of d by factoring. x^2 - 9x + 8 = 0
To solve the quadratic equation x^2 - 9x + 8 = 0 by factoring, we need to find two numbers that multiply to 8 and add to -9. These numbers are -1 and -8, since (-1)(-8) = 8 and (-1) + (-8) = -9.
So, we can rewrite the quadratic expression as:
x^2 - x - 8x + 8 = 0
Now, we can factor by grouping:
(x^2 - x) + (-8x + 8) = 0
x(x - 1) - 8(x - 1) = 0
(x - 8)(x - 1) = 0
This gives us two solutions: x = 8 or x = 1.
Therefore, the solution set is {1, 8}.
Since the question asks to solve for all values of "d", but the equation is actually in terms of "x", we can assume this is just a typo and the answer is still {1, 8}.
So, we can rewrite the quadratic expression as:
x^2 - x - 8x + 8 = 0
Now, we can factor by grouping:
(x^2 - x) + (-8x + 8) = 0
x(x - 1) - 8(x - 1) = 0
(x - 8)(x - 1) = 0
This gives us two solutions: x = 8 or x = 1.
Therefore, the solution set is {1, 8}.
Since the question asks to solve for all values of "d", but the equation is actually in terms of "x", we can assume this is just a typo and the answer is still {1, 8}.
solve the following quadratic equation for all values of x in simplest form. 3(x^2 - 2) + 5 = 5
Starting with the given equation:
3(x^2 - 2) + 5 = 5
We first simplify by subtracting 5 from both sides:
3(x^2 - 2) = 0
Next, we can divide both sides by 3:
x^2 - 2 = 0
Now, we can add 2 to both sides:
x^2 = 2
Taking the square root of both sides, we get:
x = ±√2
Therefore, the solution set is {√2, -√2}.
3(x^2 - 2) + 5 = 5
We first simplify by subtracting 5 from both sides:
3(x^2 - 2) = 0
Next, we can divide both sides by 3:
x^2 - 2 = 0
Now, we can add 2 to both sides:
x^2 = 2
Taking the square root of both sides, we get:
x = ±√2
Therefore, the solution set is {√2, -√2}.
solve the equation for all values of x by completing the square. x^2 - 2x - 48 = 0
To solve this equation by completing the square, we want to rewrite the left-hand side as a perfect square trinomial.
First, we move the constant term to the right-hand side:
x^2 - 2x = 48
Next, we need to add and subtract a constant term to complete the square. To determine which constant term to use, recall that the middle coefficient is -2. We want to find a constant c such that (c/2)^2 = c^2/4 = 1. Then we can add c^2/4 to both sides to complete the square.
To find c, we can take half of the middle coefficient and square it:
(-2/2)^2 = 1
So, we add and subtract 1:
x^2 - 2x + 1 - 1 - 48 = 0
The left-hand side is now a perfect square trinomial:
(x - 1)^2 - 49 = 0
Adding 49 to both sides:
(x - 1)^2 = 49
Taking the square root of both sides:
x - 1 = ±7
Solving for x:
x = 1 ± 7
Therefore, the solution set is {8, -6}.
First, we move the constant term to the right-hand side:
x^2 - 2x = 48
Next, we need to add and subtract a constant term to complete the square. To determine which constant term to use, recall that the middle coefficient is -2. We want to find a constant c such that (c/2)^2 = c^2/4 = 1. Then we can add c^2/4 to both sides to complete the square.
To find c, we can take half of the middle coefficient and square it:
(-2/2)^2 = 1
So, we add and subtract 1:
x^2 - 2x + 1 - 1 - 48 = 0
The left-hand side is now a perfect square trinomial:
(x - 1)^2 - 49 = 0
Adding 49 to both sides:
(x - 1)^2 = 49
Taking the square root of both sides:
x - 1 = ±7
Solving for x:
x = 1 ± 7
Therefore, the solution set is {8, -6}.
what is the discrimination of quadratic equation 4x^2 + 2x - 1 = 0
The discrimination of a quadratic equation is given by the expression b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0.
In this case, a = 4, b = 2, and c = -1.
So the discrimination is given by:
b^2 - 4ac = 2^2 - 4(4)(-1) = 20
Therefore, the discrimination of the quadratic equation 4x^2 + 2x - 1 = 0 is 20.
In this case, a = 4, b = 2, and c = -1.
So the discrimination is given by:
b^2 - 4ac = 2^2 - 4(4)(-1) = 20
Therefore, the discrimination of the quadratic equation 4x^2 + 2x - 1 = 0 is 20.