Quadratic Equations

Factorization, Quadratic formula, Completing the square, Graphical method

There are several methods to solve quadratic equations: 1) Factorization - when the equation can be factored easily, 2) Quadratic formula - works for all quadratic equations, 3) Completing the square - useful when factorization is difficult, 4) Graphical method - finding x-intercepts. Linear substitution is not a method for quadratic equations.

No real roots

When discriminant (b² - 4ac) < 0, the quadratic equation has no real roots because we cannot take the square root of a negative number in real number system. The roots are complex or imaginary.

x = 3, x = -3

Step 1: Recognize this as a difference of squares: x² - 9 = x² - 3². Step 2: Factor using a² - b² = (a + b)(a - b). Step 3: x² - 9 = (x + 3)(x - 3) = 0. Step 4: Setting each factor to zero: x + 3 = 0 or x - 3 = 0. Step 5: Therefore, x = -3 or x = 3.

4 meters

Step 1: Let width = x meters, then length = (x + 3) meters. Step 2: Area = length × width, so x(x + 3) = 28. Step 3: x² + 3x = 28, which gives x² + 3x - 28 = 0. Step 4: Factor: (x + 7)(x - 4) = 0. Step 5: x = -7 or x = 4. Step 6: Since width cannot be negative, x = 4 meters.