Let's analyze Marcus's work step-by-step.
First, we need to solve the quadratic equation using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
From the options presented, let's check their statements:
1. Marcus should have substituted -10 for [tex]$b$[/tex], not 10.
Correct. The quadratic formula requires the correct substitution of coefficients:
- Here, [tex]\( b = -10 \)[/tex].
- So, Marcus should use [tex]\( b = -10 \)[/tex] instead of [tex]\( b = 10 \)[/tex].
2. The denominator should be 1, not 2.
Incorrect. According to the quadratic formula, the denominator is [tex]\( 2a \)[/tex]:
- Here, [tex]\( a = 1 \)[/tex].
- Therefore, the denominator should be [tex]\( 2 \times 1 = 2 \)[/tex] and not 1.
3. Marcus should have subtracted [tex]\( 4(1)(25) \)[/tex] within the square root.
Correct. The discriminant of the quadratic equation is given by [tex]\( b^2 - 4ac \)[/tex]:
- Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -10 \)[/tex], and [tex]\( c = 25 \)[/tex].
- Therefore, the discriminant should be [tex]\( (-10)^2 - 4 \times 1 \times 25 \)[/tex].
Now, let's compute the discriminant and examine the number of solutions:
[tex]\[ \text{discriminant} = (-10)^2 - 4 \times 1 \times 25 = 100 - 100 = 0 \][/tex]
Since the discriminant is 0, there is exactly one solution for the quadratic equation.
4. This equation, when solved correctly, only has 1 real number solution.
Correct. A discriminant of 0 indicates that the quadratic equation has exactly one real solution.
Based on this analysis:
- Statement 1 is true.
- Statement 2 is false.
- Statement 3 is true.
- Statement 4 is true.
Therefore, the correct conclusions about Marcus's work are as follows:
- Marcus should have substituted -10 for [tex]\( b \)[/tex], not 10.
- Marcus should have subtracted [tex]\( 4(1)(25) \)[/tex] within the square root.
- This equation, when solved correctly, only has 1 real number solution.
