X²+4x-21>0

To solve the inequality x² + 4x - 21 > 0, we can use the method of factoring or the quadratic formula.

1. Factoring method: We need to factor the quadratic expression x² + 4x - 21 first: (x - 3)(x + 7) > 0

   Now, we can find the critical points by setting each factor equal to zero: x - 3 = 0 ==> x = 3 x + 7 = 0 ==> x = -7

   Since the quadratic expression is greater than zero, we need to consider the sign of each factor when x is greater than or less than the critical points:

   For x < -7, both factors are negative. So, the expression (x - 3)(x + 7) will be positive. For -7 < x < 3, the factor (x + 7) becomes positive while (x - 3) remains negative. So, the expression (x - 3)(x + 7) will be negative. For x > 3, both factors are positive. So, the expression (x - 3)(x + 7) will be positive.

   Therefore, the solution to the inequality x² + 4x - 21 > 0 is x < -7 or x > 3.

2. Quadratic formula method: We can also solve the inequality using the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a)

   For the given quadratic equation x² + 4x - 21 = 0, the coefficients are a = 1, b = 4, and c = -21.

   x = (-(4) ± √((4)² - 4(1)(-21))) / (2(1)) x = (-4 ± √(16 + 84)) / 2 x = (-4 ± √100) / 2 x = (-4 ± 10) / 2

   This gives us two values for x: x₁ = (-4 + 10) / 2 = 3 x₂ = (-4 - 10) / 2 = -7

   Since the inequality is asking for x² + 4x - 21 > 0, we only need to consider the intervals between these two values of x:

   -∞ < x < -7: Substituting a value in this interval, we get a negative result. -7 < x < 3: Substituting a value in this interval, we get a positive result. 3 < x < ∞: Substituting a value in this interval, we get a positive result.

   Therefore, the solution to the inequality x² + 4x - 21 > 0 is x < -7 or x > 3.