Write the function equation that represents the parabola with a vertex at (10, 32) and passes through the point (-2, -4) in Standard Form.
The standard form of a quadratic function equation is given by:
y = a(x - h)^2 + k
where (h, k) represents the vertex of the parabola.
In this case, the vertex is (10, 32), so h = 10 and k = 32.
We can substitute these values into the equation:
y = a(x - 10)^2 + 32
Next, we need to use the given point (-2, -4) to find the value of a.
Substituting the values of x and y into the equation, we get:
-4 = a(-2 - 10)^2 + 32
Simplifying this equation, we have:
-4 = a(-12)^2 + 32
-4 = a * 144 + 32
-4 = 144a + 32
Subtracting 32 from both sides, we get:
-36 = 144a
Dividing both sides by 144, we find:
a = -36/144
Simplifying this fraction, we get:
a = -1/4
Substituting this value of a back into the equation, we have:
y = (-1/4)(x - 10)^2 + 32
Therefore, the equation in standard form that represents the parabola is:
y = (-1/4)(x - 10)^2 + 32
The solutions of the quadratic equation 2(x - 3)^2 - 10 equals 0 are equal to the zeros of the of the graph of the corresponding quadratic function
y = 2(x - 3)^2 - 10 .
To find the zeros of the quadratic equation 2(x - 3)^2 - 10 = 0, we can set the equation equal to zero and solve for x:
2(x - 3)^2 - 10 = 0
To simplify the equation, we can first add 10 to both sides:
2(x - 3)^2 = 10
Next, divide both sides by 2:
(x - 3)^2 = 5
Now, take the square root of both sides:
x - 3 = ±√5
Finally, isolate x by adding 3 to both sides:
x = 3 ± √5
Therefore, the zeros of the quadratic function y = 2(x - 3)^2 - 10 are x = 3 + √5 and x = 3 - √5.
The equation 2x^2 + 4x + 5 = 0 has no solutions.
You are correct. The equation 2x^2 + 4x + 5 = 0 has no solutions. This can be determined using the discriminant:
The discriminant (b^2 - 4ac) for the quadratic equation ax^2 + bx + c = 0 allows us to determine the number and nature of solutions.
In this case, a = 2, b = 4, and c = 5.
Substituting these values into the discriminant formula, we have:
Discriminant = (4^2) - 4(2)(5)
Simplifying, we get:
Discriminant = 16 - 40 = -24
Since the discriminant is negative (-24 < 0), this means that the quadratic equation has no real solutions.
Use the method of your choice (factoring, Quadratic Formula or technology) to solve x^2 + 6 = 5x.
To solve the equation x^2 + 6 = 5x, we can rearrange the equation to get it in the form of a quadratic equation:
x^2 - 5x + 6 = 0
Now, we can use factoring to solve for x:
(x - 2)(x - 3) = 0
To find the values of x that satisfy the equation, we can set each factor equal to zero:
x - 2 = 0 or x - 3 = 0
Solving each equation individually:
For x - 2 = 0, we add 2 to both sides:
x = 2
For x - 3 = 0, we add 3 to both sides:
x = 3
Therefore, the solutions to the equation x^2 + 6 = 5x are x = 2 and x = 3.
Use the method of your choice to find the solution to x^2 + 10x - 25 = 3x + 9
To solve the equation x^2 + 10x - 25 = 3x + 9, we can start by rearranging the equation to get it in the form of a quadratic equation:
x^2 + 10x - 3x - 25 - 9 = 0
Simplifying, we have:
x^2 + 7x - 34 = 0
Now, we can use the quadratic formula to find the solutions for x:
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = 1, b = 7, and c = -34. Substituting these values into the quadratic formula:
x = (-7 ± √(7^2 - 4*1*(-34))) / (2*1)
Simplifying, we have:
x = (-7 ± √(49 + 136)) / 2
x = (-7 ± √185) / 2
Therefore, the solutions to the equation x^2 + 10x - 25 = 3x + 9 are:
x = (-7 + √185) / 2 and x = (-7 - √185) / 2
To write the function equation of a parabola in standard form, we use the formula:
y = a(x - h)^2 + k
Where (h, k) represents the vertex of the parabola. In this case, the vertex is (10, 32). So, h = 10 and k = 32.
Now we need to find the value of 'a' in the equation. To do this, we can substitute the coordinates (-2, -4) into the equation:
-4 = a(-2 - 10)^2 + 32
Simplifying this equation:
-4 = a(-12)^2 + 32
-4 = a(144) + 32
-4 = 144a + 32
144a = -4 - 32
144a = -36
a = -36/144
a = -1/4
Putting the values of a, h, and k into the standard form equation:
y = (-1/4)(x - 10)^2 + 32
Therefore, the function equation that represents the parabola is:
y = (-1/4)(x - 10)^2 + 32