Quadratic Inequalities
A quadratic inequality is an inequality that contains a polynomial of degree 2 and can be written in any of the following forms.
ax ^2 + bx + c > 0
ax ^2 + bx + c < 0
ax^2 + bx + c >= 0
ax^2 + bx + c <= 0
where a, b, and care real numbers and a # 0.
Examples: 1.
1.2x ^ 2 + 5x + 1 > 0
2. s ^ 2 - 9 < 2s
3. 3r ^ 2 + r - 5 >= 0
4. t ^ 2 + 4t <= 10
To solve a quadratic inequality, find the roots of its corresponding equality. The points corresponding to the roots of the equality, when plotted on the number line, separates the line into two or three intervals. An interval is part of the solution of the inequality if a number in that interval makes the inequality true.
Example 1: Find the solution set fx + 7x + 12 > 0.
The corresponding equality of 6 + 7x + 120 x ^ 2 + 7x + 12 = 0
Solve x+7x+12=0.
(x + 3)(x + 4) = 0
x + 3 = 0andx + 4 = 0
x = - 3andx = - 4
The inequality x ^ 2 + 7x + 12 > 0 is also true for values of x less than -4.
Will the inequality be true for any value of x greater than or equal to 4 but less than or equal to -32.
When ( x = - 3x ^ 2 + 7x + 12 > 0 -> (- 3) ^ 2 + 7(- 3) + 12 > 0
9 - 21 + 12 <= 0
0 > 0 (Not True)
The inequality is not true for x = - 3.
When x =-3.^ c :
x ^ 2 + 7x + 12 > 0 -> (- 3.5) ^ 2 + 7(- 3.5) + 12 > 0
12.25 - 24.5 + 12 > 0
- 0.25 > 0 (Not True)
The inequality is not true for x = - 3.5.
This shows that x ^ 2 + 7x + 12 > 0 is not true for values of x greater than or equal to -4 but less than or equal to -3.
Example 2 * 2x ^ 2 - 5x <= 3
Rewrite 2x²-5x≤ 3 to 2x-5x-3≤0. Why?
Notice that the quadratic expression 2x ^ 2 - 5x - 3 is less than or equal to zero. If we write the expression in factored form, the product of these factors must be zero or negative to satisfy the inequality. Remember that if the product of two numbers is zero, either one or both factors are zeros. Likewise, if the product of two numbers is negative, then one of these numbers is positive and the other is negative.
(2x + 1)(x - 3) <= 0
Case 1:
(2x + 1) <= 0and(x - 3) >= 0
Case 2:
(2x + 1) >= 0 and (x - 3) <= 0
Discriminant-This is the value of the expression b-4ac in the quadratic formula.
Extraneous Root or Solution - This is a solution of an equation derived from an original equation. However, it is not a solution of the original equation.
Irrational Roots-These are roots of equations which cannot be expressed as quotient of integers.
Quadratic Equations in One Variable-These are mathematical sentences of degree 2 that can be written in the form ax + bx+c=0.
Quadratic Formula-This is an equation that can be used to find the roots or solutions of the quadratic equation ax + bx + c = 0.
Quadratic Inequalities - These are mathematical sentences that can be written in any of the following forms: ax + bx + c>0, ax + bx + c <0, ax + bx + c20, and ax + bx + c≤0. Rational Algebraic Equations-These are mathematical sentences that contain rational algebraic expressions.
Rational Roots - These are roots of equations which can be expressed as quotient of integers. Solutions or Roots of Quadratic Equations- These are the values of the variable/s that make quadratic equations true.
Solutions or Roots of Quadratic Inequalities-These are the values of the variables that make quadratic inequalities true.
A quadratic inequality is an inequality that contains a polynomial of degree 2 and can be written in any of the following forms.
ax ^2 + bx + c > 0
ax ^2 + bx + c < 0
ax^2 + bx + c >= 0
ax^2 + bx + c <= 0
where a, b, and care real numbers and a # 0.
Examples: 1.
1.2x ^ 2 + 5x + 1 > 0
2. s ^ 2 - 9 < 2s
3. 3r ^ 2 + r - 5 >= 0
4. t ^ 2 + 4t <= 10
To solve a quadratic inequality, find the roots of its corresponding equality. The points corresponding to the roots of the equality, when plotted on the number line, separates the line into two or three intervals. An interval is part of the solution of the inequality if a number in that interval makes the inequality true.
Example 1: Find the solution set fx + 7x + 12 > 0.
The corresponding equality of 6 + 7x + 120 x ^ 2 + 7x + 12 = 0
Solve x+7x+12=0.
(x + 3)(x + 4) = 0
x + 3 = 0andx + 4 = 0
x = - 3andx = - 4
The inequality x ^ 2 + 7x + 12 > 0 is also true for values of x less than -4.
Will the inequality be true for any value of x greater than or equal to 4 but less than or equal to -32.
When ( x = - 3x ^ 2 + 7x + 12 > 0 -> (- 3) ^ 2 + 7(- 3) + 12 > 0
9 - 21 + 12 <= 0
0 > 0 (Not True)
The inequality is not true for x = - 3.
When x =-3.^ c :
x ^ 2 + 7x + 12 > 0 -> (- 3.5) ^ 2 + 7(- 3.5) + 12 > 0
12.25 - 24.5 + 12 > 0
- 0.25 > 0 (Not True)
The inequality is not true for x = - 3.5.
This shows that x ^ 2 + 7x + 12 > 0 is not true for values of x greater than or equal to -4 but less than or equal to -3.
Example 2 * 2x ^ 2 - 5x <= 3
Rewrite 2x²-5x≤ 3 to 2x-5x-3≤0. Why?
Notice that the quadratic expression 2x ^ 2 - 5x - 3 is less than or equal to zero. If we write the expression in factored form, the product of these factors must be zero or negative to satisfy the inequality. Remember that if the product of two numbers is zero, either one or both factors are zeros. Likewise, if the product of two numbers is negative, then one of these numbers is positive and the other is negative.
(2x + 1)(x - 3) <= 0
Case 1:
(2x + 1) <= 0and(x - 3) >= 0
Case 2:
(2x + 1) >= 0 and (x - 3) <= 0
Discriminant-This is the value of the expression b-4ac in the quadratic formula.
Extraneous Root or Solution - This is a solution of an equation derived from an original equation. However, it is not a solution of the original equation.
Irrational Roots-These are roots of equations which cannot be expressed as quotient of integers.
Quadratic Equations in One Variable-These are mathematical sentences of degree 2 that can be written in the form ax + bx+c=0.
Quadratic Formula-This is an equation that can be used to find the roots or solutions of the quadratic equation ax + bx + c = 0.
Quadratic Inequalities - These are mathematical sentences that can be written in any of the following forms: ax + bx + c>0, ax + bx + c <0, ax + bx + c20, and ax + bx + c≤0. Rational Algebraic Equations-These are mathematical sentences that contain rational algebraic expressions.
Rational Roots - These are roots of equations which can be expressed as quotient of integers. Solutions or Roots of Quadratic Equations- These are the values of the variable/s that make quadratic equations true.
Solutions or Roots of Quadratic Inequalities-These are the values of the variables that make quadratic inequalities true.
