Figure 2 (attached in the end) represents the graph of a quadratic equation with a positive discriminant.
Further explanation:
A quadratic equation is an equation in which the degree of the equation or the highest power of the variable is [tex]2[/tex].
The general representation of a quadratic equation is follows:
[tex]\fbox{\begin\\\ \math ax^{2}+bx+c=0\\\end{minispace}}[/tex]
In the above equation the terms [tex]a[/tex], [tex]b[/tex] and [tex]c[/tex] are some constants.
The curve of a quadratic equation is called a parabola.
Since, the degree of the quadratic equation is [tex]2[/tex] so, the total number of roots for the quadratic equation is [tex]2[/tex]. The two roots of the quadratic equation may be real roots or imaginary roots.
The imaginary roots of any quadratic equation always occur in their conjugate pair i.e., if one root of a quadratic equation is a complex number then the other root of the quadratic equation is the conjugate of the complex number.
The quadratic formula to obtain the roots or the zeroes of a quadratic equation is as follows:
[tex]\fbox{\begin\\\ \fbox{\begin\\\ \math x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}\\\end{minispace}}\\\end{minispace}}[/tex]
In the above equation the term [tex]b^{2}-4ac[/tex] is called the discriminant of the equation.
The nature of the roots completely depends upon the value of the discriminant.
Case 1:
Real and equal roots
If both the roots of a quadratic equation are real and equal then the curve just touches the [tex]x[/tex]-axis only at one point and that particular point is called the solution or the root of the quadratic equation.
Case 2:Imaginary roots
If the roots of a quadratic equation are imaginary then the curve of the parabola does not intersect [tex]x[/tex]-axis at any point because imaginary numbers are not plotted on the real number axis.
Case 3:
Real and unequal roots
If the roots of the quadratic equation are real and unequal then the curve of the parabola intersects the x-axis at two distinct points.
From table 1 (attached in the end) it is observed that if the value of discriminant is [tex]0[/tex] i.e., [tex]D=0[/tex] then the roots are real and equal, if [tex]D<0[/tex] then the roots are imaginary and if [tex]D>0[/tex] then the roots are real.
Since, the curve of a quadratic equation is a parabola so, if the leading coefficient is positive i.e., [tex]a>0[/tex] then the curve of the parabola is mounted upwards and if [tex]a<0[/tex] then the curve is mounted downwards.
From figure 1 (attached in the end) it is observed that if the leading coefficient is positive and the discriminant is positive i.e., the roots are real and unequal then curve is mounted upwards and intersect [tex]x[/tex]-axis at two distinct points.
If the leading coefficient is negative and the discriminant is positive then the curve is mounted downwards and the curve intersects the [tex]x[/tex]-axis at two distinct points.
Therefore, figure 1 (attached in the end) shows variation in the behavior of the curve with change in discriminant of a quadratic equation.
Learn more:
1. A problem on permutation https://brainly.com/question/5199020
2. A problem to equation of a line https://brainly.com/question/ 1575090
3. A problem to determine intercepts of a line https://brainly.com/question/1332667
Answer details:
Grade: High school
Subject: Mathematics
Chapter: Quadratic equations
Keywords: Quadratic, equation, expression, quadratic function, discriminant, imaginary, real, roots, zeroes, solution, graph, degree, highest part, complex number, quadratic formula.
