Considering the definition of zeros of the quadratic function, the dimensions of the rectangle are:
- width: 8 inches.
- length: 13 inches.
Zeros of a function
The solutions of a cuadratic function are the zeros of that 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.
In a quadratic function that has the form:
f(x)= ax² + bx + c
the zeros or roots are calculated by:
[tex]x1,x2=\frac{-b+-\sqrt{b^{2} -4ac} }{2a}[/tex]
Solutions of the equation in this case
The quadratic function is (x + 5)x = 104 or f(x) = x² + 5x – 104
Being:
the zeros or roots are calculated as:
[tex]x1=\frac{-5+\sqrt{5^{2} -4x1x(-104)} }{2x1}[/tex]
[tex]x1=\frac{-5+\sqrt{25 +416} }{2}[/tex]
[tex]x1=\frac{-5+\sqrt{441} }{2}[/tex]
[tex]x1=\frac{-5+21 }{2}[/tex]
[tex]x1=\frac{16 }{2}[/tex]
x1= 8
and
[tex]x2=\frac{-5-\sqrt{5^{2} -4x1x(-104)} }{2x1}[/tex]
[tex]x2=\frac{-5-\sqrt{25 +416} }{2}[/tex]
[tex]x2=\frac{-5-\sqrt{441} }{2}[/tex]
[tex]x2=\frac{-5-21 }{2}[/tex]
[tex]x2=\frac{-26}{2}[/tex]
x2= -13
x represents the width of the rectangle, so it cannot be a negative value.
Knowing that the rectangle has a length that is 5 inches greater than its width, the dimensions of the rectangle are:
- width: 8 inches.
- length: 13 inches.
Learn more about the zeros of a quadratic function:
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