What is a quadratic equation?

A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The highest degree of the variable x is 2.

What is the quadratic formula?

The quadratic formula is used to find the solutions of a quadratic equation and is given by: x = (-b ± √(b² - 4ac)) / 2a.

How do you determine the nature of the roots using the discriminant?

The discriminant of a quadratic equation is the value inside the square root in the quadratic formula: Δ = b² - 4ac. If Δ > 0, the equation has two real and distinct solutions. If Δ = 0, the equation has one real and repeated solution. If Δ < 0, the equation has two complex solutions.

What are the methods to solve a quadratic equation?

There are several methods to solve a quadratic equation: factoring, completing the square, using the quadratic formula, and graphing. The method chosen depends on the specific form of the quadratic equation.

Solve for x: x² - 5x + 6 = 0.

Hint: Factor the quadratic equation.

Factor the equation: (x - 2)(x - 3) = 0. Set each factor equal to zero: x - 2 = 0 or x - 3 = 0. Therefore, the solutions are x = 2 and x = 3.

Solve for x: 2x² + 3x - 2 = 0 using the quadratic formula.

Use the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a. Here, a = 2, b = 3, and c = -2. Calculate the discriminant: Δ = 3² - 4(2)(-2) = 9 + 16 = 25. Now, x = (-3 ± √25) / 4. Therefore, x = (-3 + 5) / 4 = 1/2 or x = (-3 - 5) / 4 = -2.

How do you complete the square for the equation: x² + 6x - 5 = 0?

First, move the constant to the other side: x² + 6x = 5. Then, add (6/2)² = 9 to both sides: x² + 6x + 9 = 14. Now, write the left-hand side as a perfect square: (x + 3)² = 14. Take the square root of both sides: x + 3 = ±√14. So, x = -3 ± √14.

What are the sum and product of the roots of a quadratic equation?

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