To determine the number of solutions of each quadratic equation, we will look at its discriminant. All rights reserved.\)ĭetermine the number of solutions to each quadratic equation. In the next section we will use the quadratic formula to solve quadratic equations.īiology Project > Biomath > Quadratic Functions > Roots of Quadratic EquationsĪll contents copyright © 2006. In this case, a binomial is being squared. The first step, like before, is to isolate the term that has the variable squared. Notice that the quadratic term, x, in the original form ax2 k is replaced with (x h). Thus, the graph must intersect the x-axis in two places (i.e. We can use the Square Root Property to solve an equation of the form a(x h)2 k as well. This function is graphically represented by a parabola that opens upward whose vertex lies below the x-axis. Notice that the discriminant of f( x) is greater than zero, Taking the square root of a positive real number is well defined, and the two roots are given by,Īn example of a quadratic function with two real roots is given by, If the discriminant of a quadratic function is greater than zero, that function has two real roots ( x-intercepts). Thus, the graph intersects the x-axis at exactly one point (i.e. This function is graphically represented by a parabola that opens downward and has vertex (3/2, 0), lying on the x-axis. Notice that the discriminant of f( x) is zero, The simplest example of a quadratic function that has only one real root is,Īnother example of a quadratic function with one real root is given by, Thus, a parabola has exactly one real root when the vertex of the parabola lies right on the x-axis. Notice that is the x-coordinate of the vertex of a parabola. To see this, we set b 2 −4 ac = 0 in the quadratic formula to get, If the discriminant of a quadratic function is equal to zero, that function has exactly one real root and crosses the x-axis at a single point. Thus, the graph can never intersect the x-axis and has no roots, as shown below, This function is graphically represented by a parabola that opens upward whose vertex lies above the x-axis. Notice that the discriminant of f( x) is negative, An example of a quadratic function with no real roots is given by, Since the quadratic formula requires taking the square root of the discriminant, a negative discriminant creates a problem because the square root of a negative number is not defined over the real line. If the discriminant of a quadratic function is less than zero, that function has no real roots, and the parabola it represents does not intersect the x-axis. The discriminant is important because it tells you how many roots a quadratic function has. We call the term b 2 −4 ac the discriminant. This formula is called the quadratic formula, and its derivation is included so that you can see where it comes from. Thus, the roots of a quadratic function are given by, We can do this by completing the square as, Therefore, to find the roots of a quadratic function, we set f ( x) = 0, and solve the equation, By definition, the y-coordinate of points lying on the x-axis is zero. The roots of a function are the x-intercepts. In fact, the roots of the function,Īre given by the quadratic formula. This method can be used to derive the quadratic formula, which is used to solve quadratic equations. We have already seen that completing the square is a useful method to solve quadratic equations. When we are asked to solve a quadratic equation, we are really being asked to find the roots. Therefore, a quadratic function may have one, two, or zero roots. A quadratic function is graphically represented by a parabola with vertex located at the origin, below the x-axis, or above the x-axis. Roots are also called x -intercepts or zeros. In this section, we will learn how to find the root(s) of a quadratic equation. Roots of Quadratic Equations and the Quadratic Formula Biology Project > Biomath > Quadratic Function > Roots of Quadratic Equations Quadratic Functions
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