Explanation : Therefore These two terms give you the solution. SO, work backwards. FOIL Distribute the first term to the second term. Combine like terms:. Copyright Notice. View Pre-Calculus Tutors.
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Find the Best Tutors Do not fill in this field. Your Full Name. Phone Number. Zip Code. Track your scores, create tests, and take your learning to the next level! Top Subjects. Our Company. Varsity Tutors. Privacy Policy. Therefore, a quadratic function may have one, two, or zero roots. When we are asked to solve a quadratic equation, we are really being asked to find the roots. We have already seen that completing the square is a useful method to solve quadratic equations.
This method can be used to derive the quadratic formula, which is used to solve quadratic equations. In fact, the roots of the function,. The roots of a function are the x -intercepts. By definition, the y -coordinate of points lying on the x -axis is zero. This formula is called the quadratic formula , and its derivation is included so that you can see where it comes from. The discriminant is important because it tells you how many roots a quadratic function has.
Specifically, if. One thing you might notice is that the x 2 coefficients a values are equal to 1. This simplifies the discriminant to b 2 — 4c. This guarantees that the discriminant will be positive which gives us two distinct real roots for the quadratic. One last thing you might notice is that the solutions are 1 and c where c is the constant term. After we see the pattern, we can use it to create quadratic equations with real roots. We can pick any whole number n and create a quadratic with real roots 1 and —n.
This would look like:. We can also multiply a quadratic equation by any real number except zero to get a new equation with the same roots. That means taking real numbers and writing a quadratic equation with those numbers as its solutions.
This is easy to do. Even better, there are infinitely many such equations for a given set of roots. All we need to do is FOIL x — r x — s to find the quadratic. Now you know when a quadratic equation has real roots. You also know what to look out for in terms of the discriminant, the graph, and the coefficients.
I hope you found this article helpful. If so, please share it with someone who can use the information. You can learn more about other methods for solving quadratics in this resource from Lamar University. You can learn more about quadratics in my other articles about quadratics with no real solution and quadratics with one real solution.
You might also want to read my article on when to use the quadratic equation or my article about how to factor a quadratic binomial you can use a special case of the quadratic formula. This article goes into detail on how to use a quadratic to find the nature of the solutions real or complex of a cubic function. Mean and standard deviation are often calculated and used to describe a data set, possibly to compare it to another data set. However, it helps to know the key differences between these two If you want to solve a quadratic equation, factoring is one good option.
Not only does factoring tell you the solutions, but it can make things easier if you want to graph the corresponding
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