Solve the following inequality for x:

2x² + 5x < 3

Without solving the equation 2x² + 5x = 3, how would you know that it has two rational solutions?

Calculate the percentages to find out in which assignment Tebogo performed the best. She obtained 64/75 for Assignment 1, 62/80 for Assignment 2, and 56/65 for Assignment 3.

Find the solution set of \(\sqrt{(x+1)^2 + 3x} = 0\).

The inequality 2 x 2 + 5 x < 3 can be analyzed using its discriminant to confirm it has two rational solutions. Tebogo scored approximately 86.15% in Assignment 3, making it her best performance. For the equation ( x + 1 ) 2 + 3 x ​ = 0 , the solution set is given by two real roots based on the calculated discriminant.
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Answered by Anonymous | 2025-07-04

Let's tackle each part of the problem one by one.
Part 1: Solve the inequality 2 x 2 + 5 x < 3

Rearrange the inequality to standard form: 2 x 2 + 5 x − 3 < 0

Solve the quadratic equation 2 x 2 + 5 x − 3 = 0 :



To find if this quadratic equation has two rational solutions, use the discriminant b 2 − 4 a c , where a = 2 , b = 5 , and c = − 3 .

b 2 − 4 a c = 5 2 − 4 × 2 × ( − 3 ) = 25 + 24 = 49

Since the discriminant is a perfect square ( 49 ), it has two rational solutions.



Graphical solution of inequality:


The quadratic represents a parabola that opens upwards because 0"> a = 2 > 0 .

The solution to 2 x 2 + 5 x − 3 < 0 is the interval between the two roots found from solving 2 x 2 + 5 x = 3 .


Part 2: Find which assignment Tebogo performed the best in (percentage calculations):

Assignment 1: 75 64 ​ × 100 ≈ 85.33%

Assignment 2: 80 62 ​ × 100 = 77.5%

Assignment 3: 65 56 ​ × 100 ≈ 86.15%



Tebogo performed the best in Assignment 3 with approximately 86.15%.

Part 3: Find the solution set of ( x + 1 ) 2 + 3 x ​ = 0 :

Equation: ( x + 1 ) 2 + 3 x ​ = 0

Remove the square root by squaring both sides: ( x + 1 ) 2 + 3 x = 0

Simplify and solve the quadratic equation: x 2 + 2 x + 1 + 3 x = 0 x 2 + 5 x + 1 = 0

Since the discriminant b 2 − 4 a c = 25 − 4 is not a perfect square, the equation has no real solutions.



Therefore, the solution set is empty as the expression under the square root cannot be negative.

By breaking down each part step-by-step, you can easily follow the logic and find the solutions to these mathematical problems.

Answered by AvaCharlotteMiller | 2025-07-06