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Introduction To Quadratic Equations And Inequalities

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An equation of the form ax2 + bx + c = 0 where x represents the unknown, a,b and c represents the known values and a ≠ 0. The numbers a,b and c are called the quadratic coefficients. The solution of the equation is the values which satisfy the given equation. A quadratic equation utmost has two solutions.

There are various methods to solve a quadratic equation namely:

  • Factoring by inspection
  • Completing the square

The general formula for solving a quadratic equation is

Here b2 – 4ac is called the discriminant.

A quadratic equation consists of one or two distinct real roots, or distinct complex roots.

The discriminant determines the nature of the roots.

  • The roots are distinct [both of which are real numbers] if the discriminant is positive.
  • If there is exactly one real root called a double or repeated root, then the discriminant is zero.
  • There are no real roots if the discriminant is negative. Rather there exist two distinct complex roots which are complex conjugates of one another.

Quadratic Inequalities

A mathematical statement which is identical to a quadratic equation but replaces the equal sign with inequality symbols such as greater than, lesser than is a quadratic inequality.

The quadratic inequalities are of the form ax2 + bx + c = 0 where x represents the unknown, a,b and c represents the known values and a ≠ 0.

The following are the forms of quadratic inequalities:

  • ax2 + bx + c = 0 > 0
  • ax2 + bx + c = 0 ≥ 0
  • ax2 + bx + c = 0 < 0
  • ax2 + bx + c = 0 ≤ 0

There exist infinitely many solutions, one solution or no solution for a quadratic inequality.

Steps to solve a quadratic inequality are as follows:

  • The inequality is written in the standard form with zero on the right-hand side.
  • Determine whether the graph of the quadratic function lies above or below the x-axis. If the inequality consists of “less than”, then find the x-values below the x-axis and if it is “greater than”, then find the x-values above the x-axis.
  • Quadratic inequalities are solved using a sign chart which indicates whether the function is above the x-axis using the positive signs or below the x-axis using the negative signs.
  • Find the roots of the equation and make use of the sign chart to solve the quadratic inequality.
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