Well, we know how to find the solutions using the quadratic formula :. That makes it look like we can always find solutions. However, we can't always take a square root!
Sometimes you'll get a negative number under the radical. A negative under the radical means there are no real number solutions to the radical. There are three cases:. To learn more about how to determine solutions using the x-intercepts, check out this lesson. Using the discriminant will be really helpful when you're trying to quickly solve quadratic equation word problems later on!
Let's do a little review before we investigate the discriminant. If the discriminant is less than 0, the quadratic equation has 0 real solutions. Instead of real solutions, the quadratic equation has 2 imaginary solutions. The discriminant is the part of the quadratic formula under the radical. It is used to tell you how many roots you will have. It tells us the number of real solutions for a given quadratic.
Algebra 1 Quadratic Formula and Discriminant. Go to Topic. Explanations 3. Moor Xu. The discriminant of a quadratic function is a function of its coefficients that reveals information about its roots.
Because adding and subtracting a positive number will result in different values, a positive discriminant results in two distinct solutions, and two distinct roots of the quadratic function. Since adding zero and subtracting zero in the quadratic equation lead to the same outcome, there is only one distinct root of the quadratic function.
This means the square root itself is an imaginary number, so the roots of the quadratic function are distinct and not real. Because the value is greater than 0, the function has two distinct, real zeros. Many equations with no odd-degree terms can be reduced to quadratics and solved with the same methods as quadratics. Higher degree polynomial equations can be very difficult to solve.
In some special situations, however, they can be made more manageable by reducing their exponents via substitution. If a substitution can be made such that the higher order polynomial takes the form of a quadratic, any method for solving a quadratic equation can be applied.
For example, if a quartic equation is biquadratic—that is, it includes no terms of an odd-degree— there is a quick way to find the zeroes of the quartic function by reducing it into a quadratic form.
Consider a quadratic function with no odd-degree terms which has the form:. With substitution, we were able to reduce a higher order polynomial into a quadratic equation.
It can now be solved with any of a number of methods via graphing, factoring, completing the square, or by using the quadratic formula. It is important to realize that the same kind of substitution can be done for any equation in quadratic form, not just quartics. A similar procedure can be used to solve higher-order equations. Privacy Policy. Skip to main content. Quadratic Functions and Factoring. Search for:. Introduction to Quadratic Functions.
Learning Objectives Describe the criteria for, and properties of, quadratic functions. The solutions to a quadratic equation are known as its zeros, or roots. Key Terms dependent variable : Affected by a change in input, i.
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