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Sagot :
To find the zeros of the quadratic function [tex]\( f(x) = 8x^2 - 16x - 15 \)[/tex], we follow these steps:
1. Identify the coefficients:
For the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex], we have:
[tex]\[ a = 8, \quad b = -16, \quad c = -15 \][/tex]
2. Calculate the discriminant:
The discriminant of a quadratic equation is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Plugging in the values, we get:
[tex]\[ \Delta = (-16)^2 - 4 \cdot 8 \cdot (-15) = 256 + 480 = 736 \][/tex]
3. Calculate the square root of the discriminant:
[tex]\[ \sqrt{\Delta} = \sqrt{736} \approx 27.129 \][/tex]
4. Use the quadratic formula to find the roots:
The quadratic formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substituting the known values:
[tex]\[ x = \frac{16 \pm 27.129}{16} \][/tex]
This yields two roots:
[tex]\[ x_1 = \frac{16 - 27.129}{16} \approx \frac{-11.129}{16} \approx -0.6956 \][/tex]
[tex]\[ x_2 = \frac{16 + 27.129}{16} \approx \frac{43.129}{16} \approx 2.6956 \][/tex]
Comparing the obtained roots to the provided options:
- [tex]\( x = -1 - \sqrt{2} \)[/tex] and [tex]\( x = -1 + \sqrt{2} \)[/tex]
- [tex]\( x = -1 - \sqrt{\frac{15}{8}} \)[/tex] and [tex]\( x = -1 + \sqrt{\frac{15}{8}} \)[/tex]
- [tex]\( x = 1 - \sqrt{\frac{23}{8}} \)[/tex] and [tex]\( x = 1 + \sqrt{\frac{23}{8}} \)[/tex]
- [tex]\( x = 1 - \sqrt{7} \)[/tex] and [tex]\( x = 1 + \sqrt{7} \)[/tex]
Given the numerical result of:
[tex]\(( -0.6956, 2.6956 )\)[/tex]
The correct zeros of the quadratic function [tex]\( f(x) = 8x^2 - 16x - 15 \)[/tex] are very close to:
[tex]\[ x = \frac{-11.129}{16} \approx -0.6956 \text{ and } x = \frac{43.129}{16} \approx 2.6956 \][/tex]
Thus, none of the provided symbolic options perfectly match the obtained numerical results. Therefore, from the calculated results,
[tex]\( x_1 \approx -0.6956 \)[/tex] and [tex]\( x_2 \approx 2.6956 \)[/tex] are the approximate zeros for the quadratic function [tex]\( f(x)=8x^2 - 16x - 15 \)[/tex].
1. Identify the coefficients:
For the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex], we have:
[tex]\[ a = 8, \quad b = -16, \quad c = -15 \][/tex]
2. Calculate the discriminant:
The discriminant of a quadratic equation is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Plugging in the values, we get:
[tex]\[ \Delta = (-16)^2 - 4 \cdot 8 \cdot (-15) = 256 + 480 = 736 \][/tex]
3. Calculate the square root of the discriminant:
[tex]\[ \sqrt{\Delta} = \sqrt{736} \approx 27.129 \][/tex]
4. Use the quadratic formula to find the roots:
The quadratic formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substituting the known values:
[tex]\[ x = \frac{16 \pm 27.129}{16} \][/tex]
This yields two roots:
[tex]\[ x_1 = \frac{16 - 27.129}{16} \approx \frac{-11.129}{16} \approx -0.6956 \][/tex]
[tex]\[ x_2 = \frac{16 + 27.129}{16} \approx \frac{43.129}{16} \approx 2.6956 \][/tex]
Comparing the obtained roots to the provided options:
- [tex]\( x = -1 - \sqrt{2} \)[/tex] and [tex]\( x = -1 + \sqrt{2} \)[/tex]
- [tex]\( x = -1 - \sqrt{\frac{15}{8}} \)[/tex] and [tex]\( x = -1 + \sqrt{\frac{15}{8}} \)[/tex]
- [tex]\( x = 1 - \sqrt{\frac{23}{8}} \)[/tex] and [tex]\( x = 1 + \sqrt{\frac{23}{8}} \)[/tex]
- [tex]\( x = 1 - \sqrt{7} \)[/tex] and [tex]\( x = 1 + \sqrt{7} \)[/tex]
Given the numerical result of:
[tex]\(( -0.6956, 2.6956 )\)[/tex]
The correct zeros of the quadratic function [tex]\( f(x) = 8x^2 - 16x - 15 \)[/tex] are very close to:
[tex]\[ x = \frac{-11.129}{16} \approx -0.6956 \text{ and } x = \frac{43.129}{16} \approx 2.6956 \][/tex]
Thus, none of the provided symbolic options perfectly match the obtained numerical results. Therefore, from the calculated results,
[tex]\( x_1 \approx -0.6956 \)[/tex] and [tex]\( x_2 \approx 2.6956 \)[/tex] are the approximate zeros for the quadratic function [tex]\( f(x)=8x^2 - 16x - 15 \)[/tex].
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