Respuesta :
Taking into account the discriminant and quadratic formula:
- As Δ> 0 the function has two real roots or solutions.
- The solutions are x1= 10 and x2= -2.
Zeros or solutions of a function
The points where a polynomial function crosses the axis of the independent term (x) represent the so-called zeros of the function.
That is, the zeros represent the roots of the polynomial equation that is obtained by making f(x)=0.
In summary, the roots or zeros of the quadratic function are those values of x for which the expression is equal to 0. Graphically, the roots correspond to the abscissa of the points where the parabola intersects the x-axis.
Discriminant
The function f(x) = ax² + bx + c
with a, b, c real numbers and a ≠ 0, is a function quadratic expressed in its polynomial form (It is so called because the function is expressed by a polynomial).
The following expression is called discriminant:
Δ= b²- 4×a×c
The discriminant determines the amount of roots or solutions of the function.
Then:
- If Δ <0 the function has no real roots and its graph does not intersect the x-axis.
- If Δ> 0 the function has two real roots and its graph intersects the x-axis at two points.
- If Δ = 0 the function has a real root and its graph intersects the x-axis at a single point that coincides with its vertex. In this case the function is said to have a double root.
Amount of solutions of function (x - 4)² - 28 =8
The function (x - 4)² - 28 =8 can be expressed as:
(x - 4)² - 28 -8= 0
(x - 4)² - 36= 0
x²- 8x + 16 - 36= 0
x²- 8x + 16 - 36= 0
x²- 8x - 20= 0
Being:
- a= 1
- b= -8
- c= -20
the amount of solutions are calculated as:
Δ= (-8)²- 4×1×(-20)
Δ= 144
As Δ> 0 the function has two real roots or solutions and its graph intersects the x-axis at two points.
Solutions of a cuadratic function
In a quadratic function that has the form:
f(x)= ax² + bx + c
the zeros or solutions are calculated by:
[tex]x1,x2=\frac{-b+-\sqrt{b^{2}-4ac } }{2a}[/tex]
Solutions of function (x - 4)² - 28 =8
The function (x - 4)² - 28 =8 can be expressed as x²- 8x - 20= 0
Being:
- a= 1
- b= -8
- c= -20
the solutions of the function are calculated as:
[tex]x1=\frac{-(-8)+\sqrt{(-8)^{2}-4x1x(-20) } }{2x1}[/tex]
[tex]x1=\frac{-(-8)+\sqrt{144 } }{2x1}[/tex]
[tex]x1=\frac{8+\sqrt{144 } }{2}[/tex]
[tex]x1=\frac{8+12 }{2}[/tex]
[tex]x1=\frac{20}{2}[/tex]
x1= 10
and
[tex]x2=\frac{-(-8)-\sqrt{(-8)^{2}-4x1x(-20) } }{2x1}[/tex]
[tex]x2=\frac{-(-8)-\sqrt{144 } }{2x1}[/tex]
[tex]x2=\frac{8-\sqrt{144 } }{2}[/tex]
[tex]x2=\frac{8-12}{2}[/tex]
[tex]x2=\frac{-4}{2}[/tex]
x1= -2
Finally, the quadratic function (x - 4)² - 28 =8 has two solutions and the solutions are x1= 10 and x2= -2.
Learn more about the zeros of a quadratic function:
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