Solve the inequality x^2 + 4x ≥ 77
- First step is to rearrange to make the quadratic easier to work with: in a familiar form. x2 + 4x ≥ 77, -77 from both sides x2 + 4x - 77 ≥ 0 Now the quadratic can be solved, by simplifying into double brackets.2. The x2 coefficient is 1 so this means the a*c value is -77 (from the quadratic form ax2 + bx + c) Writing down the factors of 77: 1, 77, 7, 11 < -- these factors have a difference of 4, the b value in the question.3. The equation must be 11x-7x to get +4, without jumping to the answer, write this out: x2 +11x -7x -77 ≥ 04. The first half of the equation the factor of x is taken out: x(x + 11) The second half the equation, -7 is taken out: -7(x + 11)5. Our equation is now: x(x + 11) -7(x + 11) ≥ 0 we can now effectively take a factor of (x + 11) out: (x - 7)(x + 11) ≥ 06. Sometimes you can skip steps 4 and 5 if you are used to solving quadratics. The final step of the solution is found by treating the equation as (x - 7)(x + 11) = 0. This gives the solutions x = 7 and x = -11, and we need all of the values above this, where x ≥ 7 or x ≤ -11 being the final answer.In set notation: x : x ≥ 7 or x ≤ -11 This can be hard to explain without drawing out the equation graphically, but the principle is based on the + x2 graph shape, where the further in +ve x past the highest root, the more +ve the equation becomes and the further in -ve x direction past the lowest root the more +ve the equation becomes.
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