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- https://math.libretexts.org/Courses/Highline_College/MATH_141%3A_Precalculus_I_(2nd_Edition)/03%3A_Polynomial_and_Rational_Functions/3.02%3A_Quadratic_FunctionsIf the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function. The vertex of the quadratic function is located at (\(h\), \(k\)), where ...If the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function. The vertex of the quadratic function is located at (\(h\), \(k\)), where \(h\) and \(k\) are the numbers in the transformation form of the function. The process for finding the x-intercepts is done commonly enough that sometimes people find it easier to solve the problem once in general and remember the formula for the result, rather than repeating the process each time.
- https://math.libretexts.org/Courses/Kansas_State_University/Your_Guide_to_Intermediate_Algebra/04%3A_Quadratic_and_Polynomial_Functions/4.05%3A_Solve_Quadratic_Equations_Using_the_Quadratic_FormulaWe have already seen how to solve a formula for a specific variable ‘in general’, so that we would do the algebraic steps only once, and then use the new formula to find the value of the specific vari...We have already seen how to solve a formula for a specific variable ‘in general’, so that we would do the algebraic steps only once, and then use the new formula to find the value of the specific variable. To use the Quadratic Formula, we substitute the values of \(a,b\), and \(c\) from the standard form into the expression on the right side of the formula. The solutions to a quadratic equation of the form \(a x^{2}+b x+c=0, a \neq 0\) are given by the formula:
- https://math.libretexts.org/Bookshelves/Algebra/Intermediate_Algebra_1e_(OpenStax)/09%3A_Quadratic_Equations_and_Functions/904%3A_Solve_Quadratic_Equations_Using_the_Quadratic_FormulaWhen we solved quadratic equations in the last section by completing the square, we took the same steps every time. By the end of the exercise set, you may have been wondering ‘isn’t there an easier w...When we solved quadratic equations in the last section by completing the square, we took the same steps every time. By the end of the exercise set, you may have been wondering ‘isn’t there an easier way to do this?’ The answer is ‘yes’. Mathematicians look for patterns when they do things over and over in order to make their work easier. In this section we will derive and use a formula to find the solution of a quadratic equation.
- https://math.libretexts.org/Courses/Borough_of_Manhattan_Community_College/MAT_206_Precalculus/7%3A_Trigonometric_Identities_and_Equations/7.5%3A_Solving_Trigonometric_EquationsIn earlier sections of this chapter, we looked at trigonometric identities. Identities are true for all values in the domain of the variable. In this section, we begin our study of trigonometric equat...In earlier sections of this chapter, we looked at trigonometric identities. Identities are true for all values in the domain of the variable. In this section, we begin our study of trigonometric equations to study real-world scenarios such as the finding the dimensions of the pyramids.
- https://math.libretexts.org/Bookshelves/Algebra/Intermediate_Algebra_(Arnold)/05%3A_Quadratic_Functions/5.04%3A_The_Quadratic_FormulaThe equation ax²+bx+c=0 is called a quadratic equation. Previously, we solved equations of this type by factoring and using the zero product property. It is not always possible to factor the trinomial...The equation ax²+bx+c=0 is called a quadratic equation. Previously, we solved equations of this type by factoring and using the zero product property. It is not always possible to factor the trinomial on the left-hand side of the quadratic equation as a product of factors with integer coefficients and we’ll need another method to solve the quadratic equation; the purpose of this section is to develop a formula that will consistently provide solutions of the general quadratic equation.
- https://math.libretexts.org/Courses/Northeast_Wisconsin_Technical_College/College_Technical_Math_1B_(NWTC)/04%3A_Quadratic_Equations/4.03%3A_Solve_Quadratic_Equations_Using_the_Quadratic_FormulaWe have already seen how to solve a formula for a specific variable ‘in general’ so that we would do the algebraic steps only once and then use the new formula to find the value of the specific variab...We have already seen how to solve a formula for a specific variable ‘in general’ so that we would do the algebraic steps only once and then use the new formula to find the value of the specific variable. Now, we will go through the steps of completing the square in general to solve a quadratic equation for x.
- https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_2e_(OpenStax)/03%3A_Polynomial_and_Rational_Functions/3.03%3A_Quadratic_FunctionsIn this section, we will investigate quadratic functions, which frequently model problems involving area and projectile motion. Working with quadratic functions can be less complex than working with h...In this section, we will investigate quadratic functions, which frequently model problems involving area and projectile motion. Working with quadratic functions can be less complex than working with higher degree functions, so they provide a good opportunity for a detailed study of function behavior.
- https://math.libretexts.org/Workbench/Intermediate_Algebra_2e_(OpenStax)/09%3A_Quadratic_Equations_and_Functions/9.04%3A_Solve_Quadratic_Equations_Using_the_Quadratic_FormulaWhen we solved quadratic equations in the last section by completing the square, we took the same steps every time. By the end of the exercise set, you may have been wondering ‘isn’t there an easier w...When we solved quadratic equations in the last section by completing the square, we took the same steps every time. By the end of the exercise set, you may have been wondering ‘isn’t there an easier way to do this?’ The answer is ‘yes’. Mathematicians look for patterns when they do things over and over in order to make their work easier. In this section we will derive and use a formula to find the solution of a quadratic equation.
- https://math.libretexts.org/Courses/Las_Positas_College/Foundational_Mathematics/18%3A_Quadratic_Equations_and_Functions/18.04%3A_Solve_Quadratic_Equations_Using_the_Quadratic_FormulaWhen we solved quadratic equations in the last section by completing the square, we took the same steps every time. By the end of the exercise set, you may have been wondering ‘isn’t there an easier w...When we solved quadratic equations in the last section by completing the square, we took the same steps every time. By the end of the exercise set, you may have been wondering ‘isn’t there an easier way to do this?’ The answer is ‘yes’. Mathematicians look for patterns when they do things over and over in order to make their work easier. In this section we will derive and use a formula to find the solution of a quadratic equation.
- https://math.libretexts.org/Courses/Highline_College/MATHP_141%3A_Corequisite_Precalculus/04%3A_Polynomial_and_Rational_Functions/4.02%3A_Quadratic_FunctionsIf the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function. The vertex of the quadratic function is located at (\(h\), \(k\)), where ...If the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function. The vertex of the quadratic function is located at (\(h\), \(k\)), where \(h\) and \(k\) are the numbers in the transformation form of the function. The process for finding the x-intercepts is done commonly enough that sometimes people find it easier to solve the problem once in general and remember the formula for the result, rather than repeating the process each time.
- https://math.libretexts.org/Courses/Lake_Tahoe_Community_College/A_First_Course_in_Linear_Algebra_(Kuttler)/06%3A_Complex_Numbers/6.04%3A_The_Quadratic_FormulaWhen working with real numbers, we cannot solve the quadratic formula if \(b^{2}-4ac<0.\) However, complex numbers allow us to find square roots of negative numbers, and the quadratic formula remains ...When working with real numbers, we cannot solve the quadratic formula if \(b^{2}-4ac<0.\) However, complex numbers allow us to find square roots of negative numbers, and the quadratic formula remains valid for finding roots of the corresponding quadratic equation.
