What is the solution set of the quadratic inequality
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Answer:
The solutions are (-2 , 0) and (1 , 3)
Step-by-step explanation:
∵ -x² + 4 = x + 2
* y = -x² + 4 ⇒ is quadratic represented by parabola
The parabola open downward because coefficient of x² is negative
The x-coordinate of its vertex = -b/2a, where b is the
coefficient of x and a is the coefficient of x²
∴ x = 0/2(-1) = 0
∴ The y-coordinate of the vertex = (0)² + 4 = 4
∴ The maximum point of the parabola is (0 , 4)
∵ -x² + 4 intersects x-axis at y = 0
∴ -x² + 4 = 0 ⇒ -x² = -4 ⇒ x² = 4
∵ x² = 4
∴ x = ±√4 = ± 2
∴ the parabola intersects x-axis at -2 , 2
∵ y = x + 2 represented by a line its slope = 1
It intersects y-axis at 2
∵ x + 2 intersects x-axis at y = 0
∴ x + 2 = 0
∴ x = -2
∴ The parabola and the line intersect each other at
x = -2 and y = 0
To find all the point of intersection between the 2 equations we will solve them as a system of equations
∵ y= -x² + 4 and y = x + 2
∴ -x² + 4 = x + 2
∴ -x² + 4 - x - 2 = 0
∴ -x² - x + 2 = 0 ⇒ × -1
∴ x² + x - 2 = 0 ⇒ factorize
∴ (x + 2)(x - 1) = 0
∴ x + 2 = 0 ⇒ x = -2
∴ x - 1 = 0 ⇒ x = 1
* when x = -2 ⇒ y = -2 + 2 = 0 ⇒ (-2 , 0)
* when x = 1 ⇒ y = 1 + 2 = 3 ⇒ (1 , 3)
The solution graphically
