Linear Programming Basics

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Q1. What is the vertex of the parabola represented by the equation y = x^2 - 4x + 4?

Solution:

The vertex can be found using the formula x = -b/(2a). Here, a = 1, b = -4, so x = 2. Substitute x back to find y: y = 0.

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Q2. Which of the following represents the equation of a line with a slope of -3 and y-intercept of 2?

Solution:

The slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept.

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Q3. Which of the following represents a linear equation?

Solution:

A linear equation is in the form y = mx + b. y = 2x + 3 fits this form.

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Q4. Solve the inequality 5x - 7 < 3.

Solution:

Step 1: Add 7 to both sides: 5x < 10. Step 2: Divide by 5: x < 2.

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Q5. What is the value of x in the equation 3(x - 1) = 2x + 4?

Solution:

Step 1: Distribute: 3x - 3 = 2x + 4. Step 2: Subtract 2x from both sides: x - 3 = 4. Step 3: Add 3 to both sides: x = 7.

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Q6. What is the value of x in the polynomial equation x^3 - 4x^2 + x = 0?

Solution:

Factor out x: x(x^2 - 4x + 1) = 0. Thus, x = 0 or solve x^2 - 4x + 1 = 0.

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Q7. What is the value of x in the equation 4x - 7 = 5?

Solution:

Add 7 to both sides: 4x = 12. Then divide by 4: x = 3.

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Q8. Which of the following is a polynomial?

Solution:

A polynomial is a sum of terms with non-negative integer exponents. 3x^2 + 2x - 1 fits this definition.