Question

Solve the quadratic form factor completely.

a) ax² + bx + c
b) (x + p)² + q
c) a(x - p)(x - q)
d) a(x + p)(x + q)

Answer 1

To solve a quadratic equation, one can use the quadratic formula by identifying the coefficients a, b, and c, and performing a step-by-step calculation. The roots can then be found using the quadratic formula with these specific values for a, b, and c.

Step-by-step explanation:

To solve the quadratic equation ax² + bx + c = 0, one can use the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a). This formula calculates the roots of any quadratic equation, which are the values of x that make the equation true. To find the solutions:

1. Identify the coefficients a, b, and c in the given quadratic equation.
2. Substitute these coefficients into the quadratic formula.
3. Perform the calculation step by step, starting with the discriminant (b² - 4ac), then calculating the square root, and finally solving for the two possible values of x.

Quadratic equations like (x + p)² + q and a(x - p)(x - q) are factored forms and can be expanded to their general form before applying the quadratic formula. Alternatively, if the equation is already factored as in c) a(x - p)(x - q), one can set each factor equal to zero to find the roots directly without using the quadratic formula.

For example, if the quadratic equation is x² + 0.0211x - 0.0211 = 0 and we compare it to the standard form ax² + bx + c = 0, the coefficients would be a = 1.00, b = 0.0211, and c = -0.0211. The roots can then be found using the quadratic formula with these specific values for a, b, and c.