quadratic

Examples of quadratic in the following topics:

• Solving a Quadratic Equation with the Quadratic Formula

• What is a Quadratic Function?

  • Quadratic equations are second order polynomials, and have the form $f(x)=ax^2+bx+c$.
  • A quadratic function is of the general form:
  • A quadratic equation is a specific case of a quadratic function, with the function set equal to zero:
  • All quadratic functions both increase and decrease.
  • Quadratic functions can be expressed in many different forms.

• The Quadratic Formula

  • The zeros of a quadratic equation can be found by solving the quadratic formula.
  • The quadratic formula is one tool that can be used to find the roots of a quadratic equation.  
  • The quadratic formula can always be used to find the roots of a quadratic equation, regardless of whether the roots are real or complex, whole numbers or fractions, and so on.
  • The first criterion must be satisfied to use the quadratic formula because conceptually, the formula gives the values of $x$ where the quadratic function $f(x) = ax^2+bx+c = 0$; the roots of the quadratic function.
  • Solve for the roots of a quadratic function by using the quadratic formula

• Completing the Square

  • Completing the square is a method for solving quadratic equations, and involves putting the quadratic in the form $0=a(x-h)^2 + k$.
  • Along with factoring and using the quadratic formula, completing the square is a common method for solving quadratic equations.  
  • This quadratic is not a perfect square.  
  • However, it is possible to write the original quadratic as the sum of this square and a constant:
  • Solve for the zeros of a quadratic function by completing the square

• A Graphical Interpretation of Quadratic Solutions

  • The roots of a quadratic function can be found algebraically or graphically.
  • Recall how the roots of quadratic functions can be found algebraically, using the quadratic formula $(x=\frac{-b \pm \sqrt {b^2-4ac}}{2a})$.
  • Consider the quadratic function that is graphed below.
  • Recall that the quadratic equation sets the quadratic expression equal to zero instead of $f(x)$:
  • Graph of the quadratic function $f(x) = x^2 - x - 2$

• Standard Form and Completing the Square

  • In algebra, parabolas are frequently encountered as graphs of quadratic functions, such as:
  • Completing the square may be used to solve any quadratic equation.
  • This can be applied to any quadratic equation.
  • Graph with the quadratic equation .
  • The graph of this quadratic equation is a parabola with x-intercepts at -1 and -5.

• Other Equations in Quadratic Form

  • Many equations with no odd-degree terms can be reduced to quadratics and solved with the same methods as quadratics.
  • 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.  
  • 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.  
  • Use the quadratic formula to solve any equation in quadratic form

• The Discriminant

  • where $a$, $b$ and $c$are the constants ($a$ must be non-zero) from a quadratic polynomial.
  • The discriminant $\Delta =b^2-4ac$ is the portion of the quadratic formula under the square root.
  • If ${\Delta}$ is equal to zero, the square root in the quadratic formula is zero:
  • 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.
  • Graph of a polynomial with the quadratic function $f(x) = x^2 - x - 2$.

• Linear and Quadratic Equations

  • Two kinds of equations are linear and quadratic.
  • A quadratic equation is a univariate polynomial equation of the second degree.
  • A general quadratic equation can be written in the form:
  • The term "quadratic" comes from quadratus, which is Latin for "square. " Quadratic equations can be solved by factoring, completing the square, graphing, Newton's method, and using the quadratic formula.
  • Examples of graphed quadratic equations can be seen below.

• Graphing Quadratic Equations In Standard Form

  • A quadratic function is a polynomial function of the form $y=ax^2+bx+c$.
  • Regardless of the format, the graph of a quadratic function is a parabola.
  • Each coefficient in a quadratic function in standard form has an impact on the shape and placement of the function's graph.
  • The coefficient $a$ controls the speed of increase (or decrease) of the quadratic function from the vertex.
  • Explain the meanings of the constants $a$, $b$, and $c$ for a quadratic equation in standard form