Quadratic and Logarithmic Functions

This video tutorial covers various mathematical problems, including solving quadratic equations, finding values using symmetry axes, solving inequalities, and working with exponential equations using logarithms. The instructor provides step-by-step explanations and encourages students to practice and understand the concepts for the upcoming smitb exam.

Given the quadratic equation 2x² - ax - 2 = 0, if the roots are x1 and x2, what is the value of a?

• 2

• -4

• -2

• 4

For the quadratic function f(x) = a²x² - 12x + c², if the axis of symmetry is 2/3, what is the value of a²?

• 3

• 6

• 12

• 9

In the function f(x) = a²x² - 12x + c², if the function touches the x-axis at x = 2/3, what is the value of c²?

• 2

• 4

• 8

• 6

How many negative integer solutions satisfy the inequality (x + 1)(x² + 2x - 7) ≥ x² - 1?

• 3

• 2

• 4

• 1

What is the solution to the exponential equation 2 * 5^(2x + 2) + 5^(x + 2) - 3 = 0?

• x = 1 + log5(2)

• x = 1 - log5(2)

• x = -1 + log5(2)

• x = -1 - log5(2)

If 5^x = 1/10, what is the value of x?

• -1

• -2

• 1

• 2

Given the equation 10x² - x - 12 = 0, what is the sum of all roots?

• -2

• 2

• 1

• -1

What is the value of log10 if log10 = log(p²) - log(p)?

• -1

• 1

• 2

• 0

If log(x² - x - 12) = 1, what is the value of x?

• 2

• 3

• 5

• 4

What is the simplified form of log(a * b²)?

• log(a) + 2log(b)

• 2log(a) + log(b)

• 2log(a) - log(b)

• log(a) - 2log(b)