Answer :

Answer:

Here, [tex]D>0[/tex], hence the quadratic equation has two distinct real roots.

Step-by-step explanation:

Given quadratic equation is [tex]6x^{2} +10x-1[/tex].

Let, the quadratic equation is [tex]ax^{2} +bx+c[/tex]    [where, [tex]a,b,c[/tex] are the constants]

The Discriminant [tex](D)=b^{2}-4ac[/tex]

Case [tex]1[/tex]:   [tex]b^{2}-4ac>0[/tex], if the discriminant is greater than [tex]0[/tex], it means the quadratic equation has two real distinct roots.

Case [tex]2[/tex]: [tex]b^{2}-4ac<0[/tex], if the discriminant is less than [tex]0[/tex], it means the quadratic equation has no real roots.

Case [tex]3[/tex]: [tex]b^{2}-4ac=0[/tex], if the discriminants is equal to [tex]0[/tex], it means the quadratic equation has two real identical roots.

Now,

      we have [tex]6x^{2} +10x-1[/tex], where [tex]a=6,b=10,\ and\ c=-1[/tex]

∴[tex]D=b^{2}-4ac[/tex]

     [tex]=(10)^{2}-(4\times 6\times\ -1)[/tex]

     [tex]=100+24[/tex]

     [tex]= 124[/tex]

Here, [tex]D>0[/tex], hence the quadratic equation has two distinct real roots.

Zeros of a quadratic equation are its intersection with x-axis. The considered quadratic function has 2 real and distint number of zeros.

How to use discriminant to find the property of solutions of given quadratic equation?

Let the quadratic equation given be of the form [tex]ax^2 + bx + c = 0[/tex], then

The quantity [tex]b^2 - 4ac[/tex] is called its discriminant.

The solution contains the term [tex]\sqrt{b^2 - 4ac}[/tex] which will be:

  • Real and distinct if the discriminant is positive
  • Real and equal if the discriminant is 0
  • Non-real and distinct roots if the discriminant is negative

There are two roots(also called zeros) of a quadratic equations always(assuming existence of complex numbers). We say that the considered quadratic equation has 2 solution if roots are distinct, and have 1 solutions when both roots are same.

The given quadratic function is:

[tex]f(x)=6x^2+10x-1[/tex]

To find its root, we equate the output of this equation to 0 (roots are intersection of quadratic function with x-axis(the input axis, assumingly) where the output is 0.

Thus, we get:

[tex]6x^2 + 10x - 1 = 0[/tex], whose solution will provide the roots of the quadratic function [tex]f(x)=6x^2+10x-1[/tex]

Comparing to standard form, we get:

[tex]a = 6, b= 10, c =-1[/tex]

Thus, the discriminant of this quadratic equation would be:

[tex]D = \sqrt{b^2 - 4ac} = \sqrt{10^2 - 4(6)(-1)} = \sqrt{100 +24} = \sqrt{124} > 0[/tex]

Thus, as the discriminant is positive real number, both the roots of the quadratic function [tex]f(x)=6x^2+10x-1[/tex] are real and distint.

A quadratic function can have at max 2 roots.

Thus, the quadratic function f(x)=6x^2+10x-1 has 2 real and distint number of zeros.

Learn more about discriminant of a quadratic equation here:

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