1)Write an equation for the parabola whose vertex is at (-8,4) and passes through (-6,-2)
answer=y=-3/2(x+8)^2+4
2)Solve x^2>_2x+24
answer=x<_-4 or x>_6
1)Write an equation for the parabola whose vertex is at (-8,4) and passes through (-6,-2)
answer=y=-3/2(x+8)^2+4
2)Solve x^2>_2x+24
answer=x<_-4 or x>_6
To find the equation of the parabola with a given vertex and point, you can use the vertex form of a parabola equation. The vertex form of a parabola equation is given by y = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola.
In this case, the given vertex is (-8, 4), so h = -8 and k = 4. We are also given that the parabola passes through the point (-6, -2).
To determine the value of 'a', we substitute the coordinates of the given point (-6, -2) into the equation and solve for 'a':
-2 = a(-6 - (-8))^2 + 4
-2 = a(-6 + 8)^2 + 4
-2 = a(2)^2 + 4
-2 = 4a + 4
4a = -6
a = -3/2
Substituting the values of 'h', 'k', and 'a' into the vertex form equation, we get the equation of the parabola as:
y = -3/2(x + 8)^2 + 4
Now, let's solve the inequality x^2 ≥ 2x + 24.
To solve this inequality, we first bring all terms to one side, so it becomes:
x^2 - 2x - 24 ≥ 0
Next, we factorize the quadratic equation:
(x - 6)(x + 4) ≥ 0
Now we have two factors (x - 6) and (x + 4). To determine the solutions, we consider the sign of each factor.
When the product of two factors is greater than or equal to zero, either both factors are positive or both factors are negative.
For the factor (x - 6) ≥ 0, x ≥ 6 satisfies the inequality.
For the factor (x + 4) ≥ 0, x ≥ -4 satisfies the inequality.
Therefore, the solution to the inequality x^2 ≥ 2x + 24 is:
x ≤ -4 or x ≥ 6.