What is a quadratic equation?

A quadratic equation is a polynomial equation of degree 2, typically expressed in the form ax² + bx + c = 0, where a ≠ 0, and a, b, c are constants. For example, 3x² - 2x + 5 = 0. Hint: Look for the highest exponent of the variable to confirm it's 2.

What is the discriminant and how does it determine the nature of roots?

The discriminant (D) is given by the formula D = b² - 4ac. It determines the nature of the roots: (i) D > 0 means two distinct real roots, (ii) D = 0 means one real root (a double root), and (iii) D < 0 means two complex roots. Hint: Calculate D to classify the roots.

How do you calculate the sum and product of the roots in a quadratic equation?

For a quadratic equation ax² + bx + c = 0, the sum of the roots (α + β) is -b/a and the product of the roots (αβ) is c/a. Hint: Identify the coefficients a, b, and c, and apply these formulas directly.

What is the quadratic formula to find the roots of a quadratic equation?

The quadratic formula is x = [-b ± √(b² - 4ac)] / 2a. This formula allows you to solve for x in any quadratic equation ax² + bx + c = 0. Hint: Plug in the values of a, b, and c to get the roots.

Solve for x in the equation: 2x² - 8x = 0.

Factor the equation: 2x(x - 4) = 0. Thus, x = 0 or x = 4. Hint: Factor out common terms to simplify the equation before solving.

What is the vertex form of a quadratic equation?

The vertex form of a quadratic equation is y = a(x - h)² + k, where (h, k) is the vertex of the parabola. This form is useful for graphing. Hint: Convert from standard form to vertex form to easily identify the vertex.

If the roots of a quadratic equation are 3 and -5, what is the quadratic equation?

Using the roots, the equation can be formed as x² - (3 - 5)x + (3)(-5) = 0, simplifying to x² + 2x - 15 = 0. Hint: Use the relationship between roots and coefficients to construct the equation.

What are the maximum and minimum values of a quadratic function?

The maximum or minimum value of a quadratic function occurs at x = -b/2a. If a > 0, it’s a minimum; if a < 0, it’s a maximum. Hint: Calculate -b/2a to find the x-value, then substitute it back into the function to find the value.

Find the roots of the quadratic equation x² + 4x + 4 = 0.

This factors to (x + 2)² = 0, giving the root x = -2 (a double root). Hint: Recognize perfect squares to simplify the factoring process.

What is the role of the leading coefficient in a quadratic equation?

The leading coefficient (a) in ax² + bx + c determines the direction of the parabola: if a > 0, it opens upwards; if a < 0, it opens downwards. Hint: Check the sign of a to predict the parabola's orientation.