Question

Give at least 10 quadratic inequalities in one variable in standard form

Answer 1

Answer:

Step-by-step explanation:

How to Solve Quadratic Inequalities?

A quadratic inequality is an equation of second degree that uses an inequality sign instead of an equal sign.

Examples of quadratic inequalities are: x2 – 6x – 16 ≤ 0, 2x2 – 11x + 12 > 0, x2 + 4 > 0, x2 – 3x + 2 ≤ 0 etc.

Solving a quadratic inequality in Algebra is similar to solving a quadratic equation. The only exception is that, with quadratic equations, you equate the expressions to zero, but with inequalities, you’re interested in knowing what’s on either side of the zero i.e. negatives and positives.

Quadratic equations can be solved by either the factorization method or by use of the quadratic formula. Before we can learn how to solve quadratic inequalities, let’s recall how quadratic equations are solved by handling a few examples.

How to Solve Quadratic Inequalities?

A quadratic inequality is an equation of second degree that uses an inequality sign instead of an equal sign.

Examples of quadratic inequalities are: x2 – 6x – 16 ≤ 0, 2x2 – 11x + 12 > 0, x2 + 4 > 0, x2 – 3x + 2 ≤ 0 etc.

Solving a quadratic inequality in Algebra is similar to solving a quadratic equation. The only exception is that, with quadratic equations, you equate the expressions to zero, but with inequalities, you’re interested in knowing what’s on either side of the zero i.e. negatives and positives.

Quadratic equations can be solved by either the factorization method or by use of the quadratic formula. Before we can learn how to solve quadratic inequalities, let’s recall how quadratic equations are solved by handling a few examples.

Let’s see a few examples here.

6x2– 7x + 2 = 0

Solution

⟹ 6x2 – 4x – 3x + 2 = 0

Factorize the expression;

⟹ 2x (3x – 2) – 1(3x – 2) = 0

⟹ (3x – 2) (2x – 1) = 0

⟹ 3x – 2 = 0 or 2x – 1 = 0

⟹ 3x = 2 or 2x = 1

⟹ x = 2/3 or x = 1/2

Therefore, x = 2/3, ½

Solve 3x2– 6x + 4x – 8 = 0

Solution

Factorize the expression on the left-hand side.

⟹ 3x2 – 6x + 4x – 8 = 0

⟹ 3x (x – 2) + 4(x – 2) = 0

⟹ (x – 2) (3x + 4) = 0

⟹ x – 2 = 0 or 3x + 4 = 0

⟹ x = 2 or x = -4/3

Therefore, the roots of the quadratic equation are, x = 2, -4/3.

Solve 2(x2+ 1) = 5x

Solution

2x2 + 2 = 5x

⟹ 2x2 – 5x + 2 = 0

⟹ 2x 2 – 4x – x + 2 = 0

⟹ 2x (x – 2) – 1(x – 2) = 0

⟹ (x – 2) (2x – 1) = 0

⟹ x – 2 = 0 or 2x – 1 = 0

⟹ x = 2 or x = 1/2

Therefore, the solutions are x = 2, 1/2.

(2x – 3)2= 25

Solution

Expand and factorize the expression.

(2x – 3)2 = 25

⟹ 4x2 – 12x + 9 – 25 = 0

⟹ 4x2 – 12x – 16 = 0

⟹ x2 – 3x – 4 = 0

⟹ (x – 4) (x + 1) = 0

⟹ x = 4 or x = -1

Answer 2

Quadratic Inequalities: It is almost similar to quadratic equality but instead of equality sign we put inequality sign and the equation formed is of second degree.

• $10x^2+5x+2\geq 0$
• $4x^2-x-2\leq 0$
• $x^2+x+1\geq 0$
• $6x^2-8x-9\geq 0$
• $7x^2+14x+7\leq 0$
• $3x^2+9x+12\geq 0$
• $-8x^2+4x+2\leq 0$
• $-6x^2+2x+12\leq 0$
• $2x^2-4x-16\geq 0$
• $7x^2++3x+6\leq 0$

• The above mentioned inequalities are for one variable form in second-order degree.
• Quadratic Inequalities can be formed using multiple variables.
• These quadratic inequalities will not have unique solutions.
• They might have complex solutions, two solutions, or sometimes no solution at all.
• The solution depends on the degree of the polynomial used in the inequality.
• There are several ways to solve these inequalities in the field of mathematics.