Final answer:
To show that the equation does not have any solutions, we can use an algebraic solution and a graphical representation. The algebraic solution involves solving the equation algebraically and using the discriminant to determine if it has real solutions. The graphical representation involves graphing both sides of the equation and checking if they intersect.
Step-by-step explanation:
To show that the equation does not have any solutions:
a. Algebraic solution:
We can solve the equation algebraically by combining like terms, isolating the variable on one side, and noticing that the left side of the equation does not equal the right side. Here is the step-by-step process:
- Combine like terms: 0.5x - 7 = √(-5x + 29)
- Add 7 to both sides: 0.5x = √(-5x + 36)
- Square both sides to eliminate the square root: (0.5x)² = (-5x + 36)
- Simplify: 0.25x² = -5x + 36
- Rearrange: 0.25x² + 5x - 36 = 0
- Notice that this is a quadratic equation. We can apply the discriminant to determine if it has real solutions. The discriminant is the term inside the square root of the quadratic formula: b² - 4ac
- Calculate the discriminant: b² - 4ac = (5)² - 4(0.25)(-36) = 25 - (-36) = 61
- Since the discriminant is positive, there are two real solutions
Therefore, the algebraic solution shows that the equation has solutions, contrary to the initial claim.
b. Graphical representation:
We can graph both sides of the equation and check if they intersect. If the graphs intersect, it means there is a solution. Here is the step-by-step process:
- Graph the left side of the equation: y = 0.5x - 7
- Graph the right side of the equation: y = √(-5x + 29)
- Observe the graph and check if they intersect at any point
- If the graphs do not intersect, it means there are no solutions
Therefore, the graphical representation can demonstrate that the equation does not have any solutions.