quadratic equation

Examples of quadratic equation in the following topics:

• Standard Form and Completing the Square

  • The standard form of a quadratic equation is useful for completing the square, which is used to graph the equation.
  • This form of quadratic equation is known as the "standard form" for graphing parabolas in algebra; from this equation, it is simple to determine the x-intercepts (y = 0) of the parabola, a process known as "solving" the quadratic equation.
  • Completing the square may be used to solve any quadratic equation.
  • This can be applied to any quadratic equation.
  • Graph with the quadratic equation .

• What is a Quadratic Function?

  • Quadratic equations are second order polynomials, and have the form $f(x)=ax^2+bx+c$.
  • A quadratic equation is a specific case of a quadratic function, with the function set equal to zero:
  • When all constants are known, a quadratic equation can be solved as to find a solution of $x$.  
  • Quadratic equations are different than linear functions in a few key ways.
  • is known as factored form, where $x_1$ and $x_2$ are the zeros, or roots, of the equation.

• 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.  
  • where the values of $a$, $b$, and $c$ are the values of the coefficients in the 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.
  • We can now substitute these values into the quadratic equation and simplify:

• 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.  
  • Once completing the square has been performed, the quadratic is easy to solve; because there is only one place where the variable $x$ is squared, the $(x-h)^2$ term can be isolated on one side of the equation, and then the square root of both sides can be taken.
  • This quadratic is not a perfect square.  
  • Solve for the zeros of a quadratic function by completing the square

• Factoring General Quadratics

  • We can factor quadratic equations of the form $ax^2 + bx + c$ by first finding the factors of the constant $c$.  
  • The factored form of a quadratic equation takes the general form:
  • To factor this, we have to arrange the quadratic equation in order of largest exponent value to smallest exponent value.
  • This leads to the equation:
  • Employ techniques to see whether a general quadratic equation can be factored

• 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:
  • Examples of graphed quadratic equations can be seen below.
  • Recognize the various forms in which linear and quadratic equations can be written

• 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.  
  • With substitution, we were able to reduce a higher order polynomial into a quadratic equation.  
  • It is important to realize that the same kind of substitution can be done for any equation in quadratic form, not just quartics.
  • Use the quadratic formula to solve any equation in quadratic form

• Solving Quadratic Equations by Factoring

  • A quadratic equation of the form $ax^2+bx+c=0$ can sometimes be solved by factoring the quadratic expression.
  • Factoring is useful to help solve an equation of the form:
  • For example, if you wanted to solve the equation $x^2-7x+12=0$, if you could realize that the quadratic factors as $(x-3)(x-4)$.
  • We attempt to factor the quadratic.
  • Use the factors of a quadratic equation to solve it without using the quadratic formula

• A Graphical Interpretation of Quadratic Solutions

  • 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 $f(x)$:
  • For the given equation, we have the following coefficients: $a = 1$, $b = -1$, and $c = -2$.
  • Recognize that the solutions to a quadratic equation represent where the graph of the equation crosses the x-axis

• The Discriminant

  • where $a$, $b$ and $c$are the constants ($a$ must be non-zero) from a quadratic polynomial.
  • 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$.
  • Explain how and why the discriminant can be used to find the number of real roots of a quadratic equation