Quadratic Inequalities

Equation of form ax² + bx + c = 0 where a, b and c are real numbers is called quadratic equations. If x=α is a root of the equation, then f(α)=0

The root of the quadratic equation of form ax² + bx + c = 0 is 

x = \[\frac{-b  ±  \sqrt{b^{2}-4ac}}{2a}\]

Here D is called the Discriminant of the quadratic expression and is equal to 

D = b² - 4ac 

The nature of roots depends upon the value of D 

• If D < 0, then roots are non-real and complex 

• If D > 0, then roots are real and distinct 

• If D = 0, then roots are real and equal 

Steps to Solving Quadratic Inequalities

• Factorise the quadratic equation by putting ax² + bx + c = 0 

• Determine the critical values i.e. values at which the expression becomes zero

• Plot these critical values and assign signs using the wavy curve method

• Plot a rough sketch or graph 

Solving Quadratic Inequalities

• Check the Discriminant of Quadratic Expression, f(x) = ax² + bx + c; then 

• If D > 0, then the expression can be written as f(x) = a (x – α)(x – β). where α and β are given by

• If D = 0, then the expression can be written as  f(x)=c(x−α)²

• ​If D > 0 & if A > 0, then f(x) > 0 for all ∀ X ϵ R and the expression will be cross multiplied and the sign of the inequality will not change.

• If A < 0, then f(x) < 0  for all X ϵ R and the expression will be cross multiplied and the sign of the inequality will change.

• If the expression (say ‘f’) is canceled from the same side of the inequality, then cancel it and write f ≠ 0 e.g.

Graph of Quadratic Expression

Vertex of parabola y = ax² + bx + c

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Vertex of the parabola is (-b/2a,-D/4a)

If a > 0 then parabola opens upward 

If a < 0 graph opens downward

Intersection with Axis

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1. If D > 0, parabola cuts the x-axis in two real and distinct points and opens upward if a > 0 while opening downward if a < 0

2. If D = 0 graph touches the x-axis and opens upward if a > 0 while opens downward if a < 0

3. If D < 0 graph does not touch the x-axis i.e. has no real roots and opens upward if a > 0 while opens downward if a < 0

Maximum and Minimum Values of y = ax² + bx + c 

For a > 0, the parabola y = f(x) opens upward and has minimum value at x = -b/2a 

f(-b/2a) = -D/4a

Range=(-D/4a,∞)

For a < 0, the parabola y = f(x) opens upward and has maximum value at x = -b/2a 

f(-b/2a) = -D/4a

Range=(∞,-D/4a)

Sign of Quadratic Equation

1. If b² - 4ac = 0 (double root case), then we have

ax²  + bx + c = a(x+b/2a)²

In this case, the function f(x) = ax² + bx + c has the sign of the coefficient a.

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2. If b² - 4ac > 0 (two distinct real roots case). In this case, we have

                          ax² + bx + c = a(x - x1)(x - x2)

Where x₁ and x₂ are the two roots with x₁ < x₂ Since (x - x₁)(x - x₂) is always positive when x < x₁and x > x₂, and always negative when x₁ < x< x₂ , we get

• ax² + bx + c has same sign as the coefficient a when x<x₁and x>x₂;

• ax² + bx+ c has the opposite sign as the coefficient a when x₁ < x< x₂ .

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3. If b² < 4ac (complex roots case),ax² + bx + c then has a constant sign the same as the coefficient a.

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Wavy Curve Method

• Factorize the  given polynomials

• Now make the coefficient of all the variables to factors positive

• Multiply/divide the equation both sides of the inequality by -1 remove the minus sign and by doing the inequality will reverse

• Find the roots and asymptotes of the given inequality by equating each of the factors to 0

• Now finally Plot the points on the number line and start with the largest factor. Here the curve from the positive region of the number line should intersect that point. Now let's look at the power of the factors and If it is odd, then we have to change the path of the curve from their respective roots while if it's even, continuing in the same region.

Applications of Quadratic Equations

• The golden ratio, which is useful in architectural purposes as well to study the proportions of naturally occurring objects, is found as the positive solution of the quadratic equation x2-x-1=0.

• Also the equations of geometrical shapes such as the circle and the other conic sections—ellipses, parabolas, and hyperbolas—are all quadratic equations in two variables.

• With the help of a quadratic equation, we can find the cosine or sine of the angle that is half as large as the angle already known.

• The process in which expressions are simplified employs finding the square root of the given expression, which can be done by finding the two solutions of a quadratic equation.

• Descartes' theorem also has special mentions for quadratic equations, He states that for every four mutually tangent circles, their radii satisfy a unique quadratic equation.

• Also, the equation given by Fuss' theorem, which gives us the relation between the radius of a bicentric quadrilateral's inscribed circle, the radius of its circumscribed circle, and the distance between the centers of those circles, can be demonstrated algebraically in the form of a quadratic equation for which the distance between the two circles' centers in terms of their radii is one of the solutions.

• Quadratic functions are also useful in the evaluation of the critical points of a cubic function and inflection points of a quartic function.

Avoiding Loss of Significance while solving Quadratic Inequalities

Although the quadratic formula provides an exact solution, the result is inaccurate if real numbers are approximated during the calculation stages, although it is a natural process in numerical analysis, where real numbers are approximated by floating-point numbers (called "reals" in many computer programming languages). In such cases, the quadratic formula is found to be unstable.

This error can creep in when the roots have different orders of magnitude, or, in other words, when b² and b² − 4ac are close in magnitude. In this case, if the nearly two equal numbers are deducted from each other it will cause loss of significance or catastrophic cancellation in the smaller root.

The second form of cancellation can occur when the two roots are very close, that is the roots lie between the terms b² and 4ac of the discriminant. If not careful, this causes the loss of up to half of the correct significant digits in the roots.