What is a quadratic equation?

A quadratic equation is a polynomial equation of the second degree, which can be written in the standard form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.

What are the roots of a quadratic equation?

The roots of a quadratic equation are the values of x that satisfy the equation. They can be found using the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a).

How can you determine the number of real roots a quadratic equation has?

You can determine the number of real roots using the discriminant, which is calculated as D = b² - 4ac. If D > 0, there are two distinct real roots; if D = 0, there is one real root; if D < 0, there are no real roots.

Solve for x: 2x² - 4x - 6 = 0.

Hint: Use the quadratic formula.

Identify a = 2, b = -4, and c = -6. Calculate the discriminant: D = (-4)² - 4(2)(-6) = 16 + 48 = 64. Since D > 0, there are two real roots. Now apply the quadratic formula: x = [4 ± √64] / (2*2) = [4 ± 8] / 4. This gives x = 3 and x = -1.

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

Hint: Look for two numbers that multiply to 6 and add to -5.

The numbers -2 and -3 multiply to 6 and add to -5. Therefore, the equation can be factored as (x - 2)(x - 3) = 0. The solutions are x = 2 and x = 3.

What is the vertex of the parabola represented by the quadratic equation y = ax² + bx + c?

The vertex of the parabola is given by the point (h, k), where h = -b / (2a) and k = f(h) = a(h)² + b(h) + c.

What is the axis of symmetry for the quadratic equation y = ax² + bx + c?

The axis of symmetry can be found using the formula x = -b / (2a). This is the vertical line that passes through the vertex of the parabola.

Determine the vertex of the quadratic function f(x) = 3x² - 12x + 7.

Hint: Use the vertex formula for h and then calculate f(h).

First, find h = -(-12) / (2*3) = 2. Then, calculate k = 3(2)² - 12(2) + 7 = 12 - 24 + 7 = -5. The vertex is (2, -5).

If a quadratic function opens upwards, what can you say about the coefficient a?

Hint: Think about the signs of a.

If a quadratic function opens upwards, the coefficient a is greater than 0 (a > 0). Conversely, if it opens downwards, a is less than 0 (a < 0).

Solve for x: x² + 4x + 4 = 0.

Hint: Can this be factored easily?

This can be factored as (x + 2)(x + 2) = 0, which gives the double root x = -2.