Question

Explain why the equation ( x - 4 ) ^2 - 28 =8 has two solutions. Then solve the equation to find the solutions. Show your work. ( Will Mark Brainliest). Please only answer if you are 100% sure. ​

Answer 1

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:

$x1,x2=\frac{-b+-\sqrt{b^{2}-4ac } }{2a}$

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:

$x1=\frac{-(-8)+\sqrt{(-8)^{2}-4x1x(-20) } }{2x1}$

$x1=\frac{-(-8)+\sqrt{144 } }{2x1}$

$x1=\frac{8+\sqrt{144 } }{2}$

$x1=\frac{8+12 }{2}$

$x1=\frac{20}{2}$

x1= 10

and

$x2=\frac{-(-8)-\sqrt{(-8)^{2}-4x1x(-20) } }{2x1}$

$x2=\frac{-(-8)-\sqrt{144 } }{2x1}$

$x2=\frac{8-\sqrt{144 } }{2}$

$x2=\frac{8-12}{2}$

$x2=\frac{-4}{2}$

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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