Let's analyze Marcus's work step-by-step and identify what is true about it.
The equation Marcus worked with is: \[ x^2 - 10x + 25 = 0. \]
- The correct coefficients for the quadratic formula \(-b \pm \sqrt{b^2 - 4ac} / 2a\) are:
- \(a = 1\)
- \(b = -10\) (Note that in the equation, the coefficient of \(x\) is \(-10\), not \(10\).)
- \(c = 25\)
Analyzing the options:
-
Marcus should have substituted –10 for b, not 10.
True. He mistakenly used \(10\) instead of \(-10\). -
The denominator should be 1, not 2.
False. The correct denominator is \(2a\) which is \(2 \times 1 = 2\). -
Marcus should have subtracted 4(1)(25) in the square root.
False. The correct calculation inside the square root should be \(b^2 - 4ac\) which is \( (-10)^2 - 4(1)(25) \). So it should be \(100 - 100 = 0\). -
This equation, when solved correctly, only has 1 real number solution.
True. Since the discriminant \(b^2 - 4ac = 0\), there is 1 real solution (a double root). -
This equation has 2 real number solutions.
False. As established earlier, the discrimination is zero, indicating only one real number solution.
In summary, the statements that are true concerning Marcus's work are:
- Marcus should have substituted –10 for b, not 10.
- This equation, when solved correctly, only has 1 real number solution.