Easiest Way to Solve Quadratic Inequalities

Solving quadratic inequalities problem is very easy. Seeing how hard this topic in textbooks breaks my heart. This topic is pretty simple but I just don’t understand why they are making it so complicated. This was my assigned topic for report when I was in high school and I was not able to report it in front of the class because I really don’t know how to deal with it. I asked my teacher about this and he also didn’t know how to discuss and deal with it. To make the story short, he spared me. It is also important to know how to write the intervals in symbol if you are not familiar how to use it.

quad ineq1

Forms

ax^2+bx+c>0

ax^2+bx+c\geq0

ax^2+bx+c<0

ax^2+bx+c\leq0

Let m and n are the roots of this equation since this is quadratic, the general solution set of these inequalities are the following.

Condition: m>n and a>0

Forms                                                              Solution Set
ax^2+bx+c>0                                (-\infty,n)\cup(m,+\infty)

ax^2+bx+c\geq0                           (-\infty,n]\cup[m,+\infty)

ax^2+bx+c<0                                 (n,m)

ax^2+bx+c\leq0                            [n,m]

Worked Problem 1:

Find the solution set of x^2+3x+2>0.

Solution:
Take not since a>0. We can proceed to finding the roots of this equation. Don’t mind the inequality symbol. The root of x^2-3x+2=0 is 1 and 2. The solution set is (-\infty,1)\cup(2,+\infty)

Worked Problem 2:
For what values of x is the graph of equation 36-x^2=y is below x-axis?

Solution:
The values of y that will make the graph below x-axis is negative, thus y<0.
The equation given will become 36-x^2<0. Why? Because 36-x^2 is also y.Take note now that a<0. By multiplying -1 to 36-x^2<0 1, a will become greater than 0. x^2-36>0 but the sense of inequality will also change.

The root of this equation is 6 and –6. Solution set is (-\infty,-6)\cup(6,+\infty)

Worked Problem 3:
Find the solution set of x^2-x-20\leq0.
a>0 and the root of equation is 5 and -4. It follows the fourth form of inequality. The solution set is [-4,5]

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3 Responses

  1. Paul V says:

    I would add this mnemonic, which was shown to me by a tutor: “GO L.A.”

    GO means “greater than –> ‘OR’ .” OR implies “union,” and “union” implies (left interval] union [right interval).

    L.A. means “less than –> ‘AND’ .” AND implies “intersect(ion),” and “intersect” implies just a single middle interval.

    It’s not an overwrought mmenomic and it guides the student to the end quite effectively (in my experience as a tutor). Not everyone speaks the math language, and that’s critical during the introduction to the concept. Here is where you win or lose the student.

    Best regards,…

  2. Dan Lang says:

    Great mnemonics. I can use it in the future. Thanks Paul!

  1. April 12, 2014

    […] If you don’t know how to solve this, better read first solving quadratic inequality. […]

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