step 1 in this problem the given inequality can be rewritten as x square plus x minus 2 to be less than

2-x-x^2 ≥0 | Numerade

Name: Problem 180, Chapter 7: Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry
Uploaded: 2022-01-06T02:26:59-08:00
Duration: 1 min 17 s
Channel: Aman Gupta
Description: 2-x-x^2 ≥0

Key Concepts

Graphical Interpretation of Quadratic Functions

Graphing the quadratic function provides a visual understanding of the inequality. The parabola's direction (opening upward or downward) and the points where it intersects the x-axis (the roots) help in visualizing the areas where the function is above or below the x-axis, which correspond to the regions where the inequality is satisfied.

Determining Roots using the Quadratic Formula or Factoring

To solve a quadratic inequality, the first step is often to set the inequality as an equation (ax² + bx + c = 0) to find its roots. These roots can be determined through factorization (if the quadratic is factorable) or by using the quadratic formula. The roots are significant because they partition the real number line into intervals where the expression will have constant sign.

Sign Chart / Interval Testing

Once the roots have been found, a sign chart is used to test the value of the quadratic expression in the intervals defined by these roots. By choosing test points in each interval, one can determine whether the expression is positive or negative in that interval, which ultimately helps in identifying the solution set for the inequality.

Quadratic Inequalities

Quadratic inequalities involve expressions of the form ax² + bx + c ? 0 (or similar relations) and require finding the intervals in which the quadratic expression is either non-negative or non-positive. Solving these inequalities typically involves determining the roots of the corresponding quadratic equation to split the number line into intervals, then testing the sign of the expression within those intervals.

Standard Form of Quadratic Expressions

Transforming a quadratic expression into its standard form, ax² + bx + c, is a crucial step when solving quadratic inequalities. This form allows for the systematic identification of coefficients, which is essential for applying methods like factorization, completing the square, or the quadratic formula to find the roots of the quadratic equation.

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