Math Problem Statement

F(x)=3x^2-x+7. G(x)=x^2+4x+14. How many times in the graphs of these functions intersect

Solution

To find how many times the graphs of the functions F(x)=3x2x+7F(x) = 3x^2 - x + 7 and G(x)=x2+4x+14G(x) = x^2 + 4x + 14 intersect, we need to determine when the two functions are equal. This is done by setting F(x)=G(x)F(x) = G(x) and solving for xx.

Step 1: Set the functions equal

3x2x+7=x2+4x+143x^2 - x + 7 = x^2 + 4x + 14

Step 2: Move all terms to one side

3x2x+7(x2+4x+14)=03x^2 - x + 7 - (x^2 + 4x + 14) = 0

Step 3: Simplify the equation

3x2x+7x24x14=03x^2 - x + 7 - x^2 - 4x - 14 = 0 (3x2x2)+(x4x)+(714)=0(3x^2 - x^2) + (-x - 4x) + (7 - 14) = 0 2x25x7=02x^2 - 5x - 7 = 0

Step 4: Solve the quadratic equation

We can solve the quadratic equation 2x25x7=02x^2 - 5x - 7 = 0 using the quadratic formula: x=(5)±(5)24(2)(7)2(2)x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(-7)}}{2(2)} x=5±25+564x = \frac{5 \pm \sqrt{25 + 56}}{4} x=5±814x = \frac{5 \pm \sqrt{81}}{4} x=5±94x = \frac{5 \pm 9}{4}

Step 5: Find the solutions

There are two possible solutions: x=5+94=144=3.5x = \frac{5 + 9}{4} = \frac{14}{4} = 3.5 x=594=44=1x = \frac{5 - 9}{4} = \frac{-4}{4} = -1

Step 6: Conclusion

The graphs of the functions F(x)F(x) and G(x)G(x) intersect at two points: x=3.5x = 3.5 and x=1x = -1.

Therefore, the graphs intersect two times.

Would you like further details or explanations?

Here are 5 related questions to expand on this topic:

  1. How do you find the exact points of intersection, including the yy-coordinates?
  2. Can you explain how to graph both quadratic functions and locate their intersection points visually?
  3. What is the significance of the discriminant in determining the number of intersections?
  4. How would the result change if the quadratic equation had no real solutions?
  5. Could we solve this problem by completing the square instead of using the quadratic formula?

Tip: Always check the discriminant (b24acb^2 - 4ac) when solving quadratic equations—it tells you how many real solutions exist.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Graph Intersections

Formulas

Quadratic equation: ax^2 + bx + c = 0
Quadratic formula: x = [-b ± √(b^2 - 4ac)] / 2a

Theorems

Quadratic Formula
Discriminant of a Quadratic Equation

Suitable Grade Level

Grades 9-11