What is a quadratic equation?

A quadratic equation is a polynomial equation of degree 2, typically in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.

What is the quadratic formula, and when is it used?

The quadratic formula, given by x = (-b ± √(b² - 4ac)) / (2a), is used to find the solutions (roots) of a quadratic equation when factoring is difficult or impossible.

Solve the quadratic equation: x² - 5x + 6 = 0.

Hint: Try factoring first.

Factoring the equation gives (x - 2)(x - 3) = 0. Setting each factor to zero gives the solutions: x = 2 and x = 3.

What does the discriminant tell you about the roots of a quadratic equation?

The discriminant, calculated as D = b² - 4ac, indicates the nature of the roots: If D > 0, there are two distinct real roots; if D = 0, there is one real root (a repeated root); if D < 0, there are no real roots (the roots are complex).

Given the quadratic equation 2x² + 4x + 2 = 0, use the quadratic formula to find the solutions.

Hint: Identify a, b, and c first.

Here, a = 2, b = 4, and c = 2. The discriminant is D = 4² - 4(2)(2) = 16 - 16 = 0. Using the quadratic formula: x = (-4 ± √0) / (2*2) = -4/4 = -1. There is one solution: x = -1.

What is the vertex form of a quadratic equation, and how is it derived?

The vertex form of a quadratic equation is y = a(x - h)² + k, where (h, k) is the vertex of the parabola. It is derived by completing the square on the standard form ax² + bx + c.

How can you determine the axis of symmetry of a quadratic function?

The axis of symmetry for the quadratic function y = ax² + bx + c is given by the formula x = -b / (2a). This line divides the parabola into two mirror-image halves.

If the quadratic equation x² - 6x + k = 0 has exactly one solution, what is the value of k?

Hint: Use the discriminant.

For the equation to have exactly one solution, the discriminant must be zero. Thus, D = b² - 4ac = (-6)² - 4(1)(k) = 36 - 4k = 0. Solving gives k = 9.

What is the relationship between the roots of a quadratic equation and its coefficients?

For a quadratic equation ax² + bx + c = 0, the sum of the roots (r₁ + r₂) is given by -b/a, and the product of the roots (r₁r₂) is given by c/a.

Find the roots of the quadratic equation 3x² - 12x + 9 = 0 using the quadratic formula.

Hint: Calculate the discriminant first.

Here, a = 3, b = -12, and c = 9. The discriminant is D = (-12)² - 4(3)(9) = 144 - 108 = 36. Using the quadratic formula: x = (12 ± √36) / (2*3) = (12 ± 6) / 6, resulting in x = 3 and x = 1.

What is the effect of changing the coefficient 'a' in the quadratic equation ax² + bx + c?

Changing the coefficient 'a' affects the width and direction of the parabola. If a > 0, the parabola opens upwards; if a < 0, it opens downwards. A larger |a| makes the parabola narrower.