University of Kwazulu-Natal

Quadratic Inequalities Definition Expression Graphs Solved Examples

Bonisiwe Shabane
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quadratic inequalities definition expression graphs solved examples

Quadratic inequalities can be derived from quadratic equations. The word “quadratic” comes from the word “quadrature”, which means "square" in Latin. From this, we can define quadratic inequalities as second-degree inequation. Here, we first define a quadratic equation. The general form of a quadratic equation is ax2 + bx + c = 0. Further if the quadratic polynomial ax2 + bx + c is not equal to zero, then they are either ax2 + bx + c > 0, or ax2 + bx + c < 0,...

In this mini-lesson, we will learn about quadratic inequality, solving quadratic inequalities, quadratic inequalities in one variable, quadratic inequalities formula, and the graph of quadratic inequality, with the help of examples, FAQs. The quadratic inequality is a second-degree expression in x and has a greater than (>) or lesser than (<) inequality. the quadratic inequality has been derived from the quadratic equation ax2 + bx + c = 0. Let us check the definition of quadratic inequality, the standard form, and the examples of quadratic inequalities. If a quadratic polynomial in one variable is less than or greater than some number or any other polynomial (with a degree less than or equal to 2), then it is said to be... The difference between a quadratic equation and a quadratic inequality is that the quadratic equation is equal to some number while quadratic inequality is either less than or greater than some number.

Some examples of quadratic inequalities in one variable are: The standard form of quadratic inequalities in one variable is almost the same as the standard form of a quadratic equation. The only difference is that the quadratic equation has an "equal to" sign in it while a quadratic inequality has a "greater than" or "less than" sign (> or <). The standard form of quadratic inequality can be represented as: A Quadratic Equation (in Standard Form) looks like: A Quadratic Equation in Standard Form (a, b, and c can have any value, except that a can't be 0.)

The above is an equation (=) but sometimes we need to solve inequalities like these: Solving inequalities is very like solving equations ... we do most of the same things. x2 − x − 6 has these simple factors (what luck!): Home » Algebra » Inequalities » Quadratic Inequalities If we replace a quadratic equation’s equality sign (=) in the standard form ax2 + bx + c = 0 with an inequality sign, it becomes a quadratic inequality.

Here are a few examples of quadratic inequalities: Depending on the sign, the 4 standard forms of quadratic inequality are: Here, like a quadratic equation, ‘a’ (≠ 0), ‘b,’ and ‘c’ are the constants, and ‘x’ is a variable. Like equations have different forms, inequalities also exist in different forms, and quadratic inequality is one of them. A quadratic inequality is an equation of second degree that uses an inequality sign instead of an equal sign. The solutions to quadratic inequality always give the two roots.

The nature of the roots may differ and can be determined by discriminant (b2 – 4ac). The general forms of the quadratic inequalities are: x2 – 6x – 16 ≤ 0, 2x2 – 11x + 12 > 0, x2 + 4 > 0, x2 – 3x + 2 ≤ 0 etc. Our chosen students improved 1.19 of a grade on average - 0.45 more than those who didn't have the tutoring. In order to access this I need to be confident with: Here we will learn about quadratic inequalities including how to solve quadratic inequalities, identify solution sets using inequality notation and represent solutions on a number line.

There are also quadratic inequalities worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck. Quadratic inequalities are similar to quadratic equations and when plotted they display a parabola. We can solve quadratic inequalities to give a range of solutions. Quadratic inequalities are an extension of quadratic equations and play a vital role in algebra. Unlike equations that ask for specific values where an expression equals zero, quadratic inequalities are concerned with ranges of values that make the expression greater than or less than zero (or equal to it). Understanding quadratic inequalities is easier—and more intuitive—when you pair algebraic steps with graphical interpretation.

In this article, we’ll break down what quadratic inequalities are and explain them using clear graphs and real-life contexts. A quadratic inequality is any inequality that contains a quadratic expression, usually in one of these forms: Here, a, b, and c are constants, and x is the variable. The highest degree of x is 2, which means the graph of a quadratic expression is a parabola. The graph of a quadratic expression y = ax² + bx + c is a parabola that: The table below represents two general formulas that express the solution of a quadratic inequality of a parabola that opens upwards (ie a > 0) whose roots are r1 and r2.

We can reproduce these general formula for inequalities that include the quadratic itself (ie ≥ and ≤). Warning about imaginary solutions: Although the solution of a quadratic equation could be imaginary. The solution of a quadratic inequality cannot include imaginary numbers -- this is because imaginary numbers cannot be ordered. How to graph and solve a quadratic inequality A quadratic inequality compares a quadratic expression ax² + bx + c (where a ≠ 0) to zero using >, <, ≥, ≤ signs. They allow us to analyze intervals between roots and help determine if a given condition holds.

Students should rewrite the inequality in standard form for solving, find the roots of ax² + bx + c = 0, and test sign intervals. A quadratic inequality involves comparing a quadratic expression \(ax^2 + bx + c\), where \(a ≠ 0 \) to a number or another polynomial (degree ≤ 2), using >, <, ≥, or ≤. Unlike a quadratic equation (which equals something), inequalities yield ranges of solutions. Examples are: Quadratic inequalities compare a quadratic expression to zero, using inequality symbols: greater than, less than, greater than or equal to, and less than or equal to. They are solved by finding roots and testing the signs of the expression over different intervals.

Step 1 - Rewrite the equation to express the inequality: Example: \(x^2 − 5x + 6 > 0\) Step 2 - Find roots by factoring or formula: \((x−2)(x−3)>0\) → roots: 2, 3. Home Linear Equations and Inequalitie Solving Inequalities Absolute Value Inequalities Graphing Equivalent Fractions Lesson Plan Investigating Liner Equations Using Graphing Calculator Graphically solving a System of two Linear Equatio Shifting Reflecting Sketching Graph Graphs... Please use this form if you would like to have this math solver on your website, free of charge. You can use the graph of a quadratic function to solve quadratic inequalities. The original inequality asks for the values of x for which the parabola is below the x-axis:

The parabola is below the x-axis for −1 < x < 3.

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