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Sagot :
Sure, let's solve the quadratic equation [tex]\(4x^2 + 45x + 24 = 0\)[/tex] using the quadratic formula step by step.
The quadratic formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, the coefficients are:
- [tex]\( a = 4 \)[/tex]
- [tex]\( b = 45 \)[/tex]
- [tex]\( c = 24 \)[/tex]
1. Calculate the Discriminant:
The discriminant [tex]\(\Delta\)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Plugging in the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[ \Delta = 45^2 - 4 \cdot 4 \cdot 24 \][/tex]
First, calculate [tex]\( 45^2 \)[/tex]:
[tex]\[ 45^2 = 2025 \][/tex]
Then, calculate [tex]\( 4 \cdot 4 \cdot 24 \)[/tex]:
[tex]\[ 4 \cdot 4 \cdot 24 = 384 \][/tex]
So, the discriminant is:
[tex]\[ \Delta = 2025 - 384 = 1641 \][/tex]
2. Calculate the Two Solutions:
Using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
First, calculate [tex]\( \sqrt{1641} \)[/tex]. Since the exact square root value is not a standard number, we'll use its approximate value in the result, knowing that it can be obtained through calculation methods (e.g., using a calculator).
Let's denote [tex]\(\sqrt{1641}\)[/tex] with a symbol for ease, suppose [tex]\(\sqrt{1641} \approx 40.533\)[/tex] (though we know this symbolically, so keep in reference to your earlier numeric results).
Now calculate the two solutions:
[tex]\[ x_1 = \frac{-45 + 40.533}{2 \cdot 4} \][/tex]
[tex]\[ x_2 = \frac{-45 - 40.533}{2 \cdot 4} \][/tex]
For [tex]\( x_1 \)[/tex]:
[tex]\[ x_1 = \frac{-45 + 40.533}{8} = \frac{-4.467}{8} \approx -0.558375 \][/tex]
For [tex]\( x_2 \)[/tex]:
[tex]\[ x_2 = \frac{-45 - 40.533}{8} = \frac{-85.533}{8} \approx -10.691625 \][/tex]
Alternatively approximated (more precision match):
[tex]\[ x_1 \approx -0.56134 \][/tex]
[tex]\[ x_2 \approx -10.68866 \][/tex]
So, the solutions to the quadratic equation [tex]\( 4x^2 + 45x + 24 = 0 \)[/tex] are approximately:
- [tex]\( x_1 \approx -0.56134 \)[/tex] (So, closest choice-wise: [tex]\(-0.56\)[/tex])
- [tex]\( x_2 \approx -10.68866\)[/tex] (So, closest choice-wise: [tex]\(-10.69\)[/tex])
None of the incorrect options provided match directly, and with quadratic roots, we are aware to be precise rather closer to decimal outputs.
Thus, final selected roots for given are:
[tex]\[ \boxed{\left( \frac{-45 + \sqrt{1641}}{8}, \frac{-45 - \sqrt{1641}}{8} \right) } = (-0.56134, -10.68866) \][/tex]
Correct solution interpretation given answers close - follow it:
- First close root option: [tex]\(-0.56\)[/tex]
- Second close root option: Eventually matches
Other choice closer remains needed - selection close enough to guidance give correctly to verify).
The quadratic formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, the coefficients are:
- [tex]\( a = 4 \)[/tex]
- [tex]\( b = 45 \)[/tex]
- [tex]\( c = 24 \)[/tex]
1. Calculate the Discriminant:
The discriminant [tex]\(\Delta\)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Plugging in the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[ \Delta = 45^2 - 4 \cdot 4 \cdot 24 \][/tex]
First, calculate [tex]\( 45^2 \)[/tex]:
[tex]\[ 45^2 = 2025 \][/tex]
Then, calculate [tex]\( 4 \cdot 4 \cdot 24 \)[/tex]:
[tex]\[ 4 \cdot 4 \cdot 24 = 384 \][/tex]
So, the discriminant is:
[tex]\[ \Delta = 2025 - 384 = 1641 \][/tex]
2. Calculate the Two Solutions:
Using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
First, calculate [tex]\( \sqrt{1641} \)[/tex]. Since the exact square root value is not a standard number, we'll use its approximate value in the result, knowing that it can be obtained through calculation methods (e.g., using a calculator).
Let's denote [tex]\(\sqrt{1641}\)[/tex] with a symbol for ease, suppose [tex]\(\sqrt{1641} \approx 40.533\)[/tex] (though we know this symbolically, so keep in reference to your earlier numeric results).
Now calculate the two solutions:
[tex]\[ x_1 = \frac{-45 + 40.533}{2 \cdot 4} \][/tex]
[tex]\[ x_2 = \frac{-45 - 40.533}{2 \cdot 4} \][/tex]
For [tex]\( x_1 \)[/tex]:
[tex]\[ x_1 = \frac{-45 + 40.533}{8} = \frac{-4.467}{8} \approx -0.558375 \][/tex]
For [tex]\( x_2 \)[/tex]:
[tex]\[ x_2 = \frac{-45 - 40.533}{8} = \frac{-85.533}{8} \approx -10.691625 \][/tex]
Alternatively approximated (more precision match):
[tex]\[ x_1 \approx -0.56134 \][/tex]
[tex]\[ x_2 \approx -10.68866 \][/tex]
So, the solutions to the quadratic equation [tex]\( 4x^2 + 45x + 24 = 0 \)[/tex] are approximately:
- [tex]\( x_1 \approx -0.56134 \)[/tex] (So, closest choice-wise: [tex]\(-0.56\)[/tex])
- [tex]\( x_2 \approx -10.68866\)[/tex] (So, closest choice-wise: [tex]\(-10.69\)[/tex])
None of the incorrect options provided match directly, and with quadratic roots, we are aware to be precise rather closer to decimal outputs.
Thus, final selected roots for given are:
[tex]\[ \boxed{\left( \frac{-45 + \sqrt{1641}}{8}, \frac{-45 - \sqrt{1641}}{8} \right) } = (-0.56134, -10.68866) \][/tex]
Correct solution interpretation given answers close - follow it:
- First close root option: [tex]\(-0.56\)[/tex]
- Second close root option: Eventually matches
Other choice closer remains needed - selection close enough to guidance give correctly to verify).
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