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Sagot :
Let's analyze Marcus's work step-by-step and identify the correctness of his calculations and substitutions based on the quadratic equation [tex]\( x^2 - 10x + 25 = 0 \)[/tex].
1. Quadratic Formula:
The quadratic formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
For the quadratic equation [tex]\( x^2 - 10x + 25 = 0 \)[/tex], the coefficients are:
[tex]\( a = 1 \)[/tex], [tex]\( b = -10 \)[/tex], and [tex]\( c = 25 \)[/tex].
2. Analyzing Marcus's Work:
- First Statement in Marcus's Work:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
This is the correct quadratic formula.
- Second Statement in Marcus's Work:
[tex]\[ x = \frac{-(10) \pm \sqrt{(10)^2 + 4(1)(25)}}{2(1)} \][/tex]
Here, Marcus substituted [tex]\( b = 10 \)[/tex] instead of [tex]\( b = -10 \)[/tex]. The correct substitution should have been:
[tex]\[ x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(1)(25)}}{2(1)} \][/tex]
So, Marcus should have substituted [tex]\(-10\)[/tex] for [tex]\( b \)[/tex], not [tex]\( 10 \)[/tex].
- Third Statement in Marcus's Work:
[tex]\[ x = \frac{-10 \pm \sqrt{200}}{2} \][/tex]
Marcus calculated the denominator incorrectly as [tex]\( 1 \)[/tex] instead of [tex]\( 2 \)[/tex]. The correct denominator should be [tex]\( 2 \cdot 1 = 2 \)[/tex].
Also, inside the square root, he incorrectly added [tex]\( 4ac \)[/tex] instead of subtracting it:
[tex]\[ b^2 - 4ac = (-10)^2 - 4(1)(25) = 100 - 100 = 0 \][/tex]
Therefore, Marcus should have subtracted [tex]\( 4(1)(25) \)[/tex] in the square root.
3. Number of Real Solutions:
To determine the number of real solutions, we check the discriminant [tex]\( b^2 - 4ac \)[/tex]:
[tex]\[ (-10)^2 - 4(1)(25) = 0 \][/tex]
When the discriminant is zero, the quadratic equation has exactly one real solution. Hence, this equation, when solved correctly, has only one real number solution.
4. Summary of Correct Statements:
- Marcus should have substituted [tex]\(-10\)[/tex] for [tex]\( b \)[/tex], not [tex]\( 10 \)[/tex].
- The denominator should be [tex]\( 2 \)[/tex], not [tex]\( 1 \)[/tex].
- Marcus should have subtracted [tex]\( 4(1)(25) \)[/tex] in the square root.
- This equation, when solved correctly, only has one real number solution.
- This equation does not have two real number solutions.
Therefore, the correct statements about Marcus's work are:
- Marcus should have substituted [tex]\(-10\)[/tex] for [tex]\( b \)[/tex], not [tex]\( 10 \)[/tex].
- Marcus should have subtracted [tex]\( 4(1)(25) \)[/tex] in the square root.
- This equation, when solved correctly, only has one real number solution.
Thus, the correct responses are:
- Marcus should have substituted -10 for [tex]\( b \)[/tex], not 10 [tex]\( \rightarrow \text{True} \)[/tex]
- The denominator should be 1, not 2 [tex]\( \rightarrow \text{False} \)[/tex]
- Marcus should have subtracted [tex]\( 4(1)(25) \)[/tex] in the square root [tex]\( \rightarrow \text{True} \)[/tex]
- This equation, when solved correctly, only has 1 real number solution [tex]\( \rightarrow \text{True} \)[/tex]
- This equation has 2 real number solutions [tex]\( \rightarrow \text{False} \)[/tex]
1. Quadratic Formula:
The quadratic formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
For the quadratic equation [tex]\( x^2 - 10x + 25 = 0 \)[/tex], the coefficients are:
[tex]\( a = 1 \)[/tex], [tex]\( b = -10 \)[/tex], and [tex]\( c = 25 \)[/tex].
2. Analyzing Marcus's Work:
- First Statement in Marcus's Work:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
This is the correct quadratic formula.
- Second Statement in Marcus's Work:
[tex]\[ x = \frac{-(10) \pm \sqrt{(10)^2 + 4(1)(25)}}{2(1)} \][/tex]
Here, Marcus substituted [tex]\( b = 10 \)[/tex] instead of [tex]\( b = -10 \)[/tex]. The correct substitution should have been:
[tex]\[ x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(1)(25)}}{2(1)} \][/tex]
So, Marcus should have substituted [tex]\(-10\)[/tex] for [tex]\( b \)[/tex], not [tex]\( 10 \)[/tex].
- Third Statement in Marcus's Work:
[tex]\[ x = \frac{-10 \pm \sqrt{200}}{2} \][/tex]
Marcus calculated the denominator incorrectly as [tex]\( 1 \)[/tex] instead of [tex]\( 2 \)[/tex]. The correct denominator should be [tex]\( 2 \cdot 1 = 2 \)[/tex].
Also, inside the square root, he incorrectly added [tex]\( 4ac \)[/tex] instead of subtracting it:
[tex]\[ b^2 - 4ac = (-10)^2 - 4(1)(25) = 100 - 100 = 0 \][/tex]
Therefore, Marcus should have subtracted [tex]\( 4(1)(25) \)[/tex] in the square root.
3. Number of Real Solutions:
To determine the number of real solutions, we check the discriminant [tex]\( b^2 - 4ac \)[/tex]:
[tex]\[ (-10)^2 - 4(1)(25) = 0 \][/tex]
When the discriminant is zero, the quadratic equation has exactly one real solution. Hence, this equation, when solved correctly, has only one real number solution.
4. Summary of Correct Statements:
- Marcus should have substituted [tex]\(-10\)[/tex] for [tex]\( b \)[/tex], not [tex]\( 10 \)[/tex].
- The denominator should be [tex]\( 2 \)[/tex], not [tex]\( 1 \)[/tex].
- Marcus should have subtracted [tex]\( 4(1)(25) \)[/tex] in the square root.
- This equation, when solved correctly, only has one real number solution.
- This equation does not have two real number solutions.
Therefore, the correct statements about Marcus's work are:
- Marcus should have substituted [tex]\(-10\)[/tex] for [tex]\( b \)[/tex], not [tex]\( 10 \)[/tex].
- Marcus should have subtracted [tex]\( 4(1)(25) \)[/tex] in the square root.
- This equation, when solved correctly, only has one real number solution.
Thus, the correct responses are:
- Marcus should have substituted -10 for [tex]\( b \)[/tex], not 10 [tex]\( \rightarrow \text{True} \)[/tex]
- The denominator should be 1, not 2 [tex]\( \rightarrow \text{False} \)[/tex]
- Marcus should have subtracted [tex]\( 4(1)(25) \)[/tex] in the square root [tex]\( \rightarrow \text{True} \)[/tex]
- This equation, when solved correctly, only has 1 real number solution [tex]\( \rightarrow \text{True} \)[/tex]
- This equation has 2 real number solutions [tex]\( \rightarrow \text{False} \)[/tex]
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