simplify the expression to a polynomial in standard form: (x + 5)(2x^2 + 2x - 3)
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.
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).
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}.
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}.
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}.