# How To: Solve quadratic inequalities in calculus

Calculus can be a bit tricky, but don't worry, Watch Math is here to help you out. Watch this math video lesson to learn how to solve quadratic inequalities in calculus. There's no better way to learn mathematics than from an advanced mathematician, like this PhD.

The trick to solving a quadratic inequality is to replace the inequality symbol with an equal sign and solve the resulting equation. The solutions to the equation will allow you to establish intervals that will let you solve the inequality.

Plot the solutions on a number line creating the intervals for investigation. Pick a number from each interval and test it in the original inequality. If the result is true, that interval is a solution to the inequality.

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Solving quadratic inequalities by substitution the values of x into the inequality takes lot of time. There is another worldwide concept in solving quadratic inequalities that bases on application of a theorem about trinomials. Students can study one time the theorem then they can apply it to solving all quadratic inequalities. See book titled: New methods for solving quadratic equations and inequalities" (Trafford 2009)

How many methods are there to solve quadratic inequalities ?
There are 3 methods to solve a quadratic inequality in standard form: ax^2 + bx + c 0 (or

There are 3 methods for solving quadratic inequalities in standard form
1 Solving by the number line
2. Solving by the algebraic method
3. Solving by graphing

There are 3 steps in solving quadratic inequalities:
1. Transform the left side of the inequality into a trinomial f (x) in x
2. Solve the quadratic equation f(x) = 0
3. Solve the inequalities by substitution with x = 0 in the number-line method or by using the theorem about the sign of the trinomial f(x)

Answers (solution set) in solving quadratic inequalities are usually given in intervals. Students must be able to express answers in the form of intervals: Exp (3, 7) ; (3, 7] ; [3, 7] ; {3, 7) ; {3, 7} ; (-5/3, 4/9]
Students must also know the notion and symbol of the 2 infinities since they are part of the solution set

Theorem on the sign status of a trinomial f(x);
1. If D = b^2 - 4ac 0 Discriminant D

1. If D = b^2 - 4ac is negative. There are no real roots; the trinomial f(x) has the sign of a for all values of x. Exp: f(x) = x^2 - 2x + 7 0; D is neg., f(x) is always positive
2. If D is positive: There are two real roots x1 and x2. Between the real roots f(x) has the opposite sign of a. Exp: f(x) = x^2 - 4x + 3

1. If D = b^2 - 4ac 0: D 0: f(x) has opposite sign of a between the 2 real roots x1 and x2
Exp: f(x) = x^2 - 3x + 2

3. Solve the quadratic inequality. There are two approaches:
a. Use substitution method (point checking). I advise you to always select x = 0
b. Use the theorem about the sign of a trinomial f(x)

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