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Answer:

A

Step-by-step explanation:

CintiaSalazar

Considering the definition of discriminant and the expression to find the solution of a quadratic equation:

  • the graph of an equation with a negative discriminant always has no x-intercept.
  • the solutions of the function 5x = 6x² – 3 are [tex]x1=\frac{5+\sqrt{97} }{12}[/tex] and [tex]x2=\frac{5-\sqrt{97} }{12}[/tex] or ,which is the same, [tex]x1,x2=\frac{5+-\sqrt{97} }{12}[/tex]

The general form of a quadratic equation is ax² +bx +c, where a≠0

To find the solution of a quadratic equation, the expression is used:

[tex]x1,x2=\frac{-b+-\sqrt{b^{2}-4ac } }{2a}[/tex]

Discriminant

The part that is affected by the square root is called discriminant and is denoted by Δ:

Δ=b²-4ac

The sign of the discriminant informs the number of solutions of the equation:

  • If Δ is positive (Δ> 0), the equation has two different solutions.
  • If Δ is null (Δ = 0), the equation has only one solution (that is, it has two equal solutions).
  • If Δ is negative (Δ <0), the equation has no real solutions (it has two complex or imaginary solutions).

In other words:

  • If Δ> 0, then the parabola intersects the x-axis at two points.
  • If Δ = 0, then the parabola intersects the x-axis at one point.
  • If Δ <0, then the parabola does not intersect the x-axis.

In summary, the graph of an equation with a negative discriminant always has no x-intercept.

Solve 5x = 6x² – 3

On the other side, you have to solve 5x = 6x² – 3

Expressing as the general form for the quadratic function you have:

6x² -5x – 3= 0

where a=6, b=-5 and c=-3

The solution of the quadratic equation can be calculated as:

[tex]x1,x2=\frac{-(-5)+-\sqrt{(-5)^{2}-4x6x(-3)} }{2x6}[/tex]

Solving:

[tex]x1=\frac{-(-5)+\sqrt{(-5)^{2}-4x6x(-3)} }{2x6}[/tex]

[tex]x1=\frac{5+\sqrt{25+72} }{12}[/tex]

[tex]x1=\frac{5+\sqrt{97} }{12}[/tex]

and

[tex]x2=\frac{-(-5)-\sqrt{(-5)^{2}-4x6x(-3)} }{2x6}[/tex]

[tex]x2=\frac{5-\sqrt{25+72} }{12}[/tex]

[tex]x2=\frac{5-\sqrt{97} }{12}[/tex]

So, the solutions of the function 5x = 6x² – 3 are [tex]x1=\frac{5+\sqrt{97} }{12}[/tex] and [tex]x2=\frac{5-\sqrt{97} }{12}[/tex] or ,which is the same, [tex]x1,x2=\frac{5+-\sqrt{97} }{12}[/tex]

Learn more about the discriminant and the expression to find the solution of a quadratic equation:

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