Write it out. Your tutor reads along.

The problem: For which values of k does x² − kx + k + 3 = 0 have two distinct real roots?

The student writes on the page.

The tutor underlines a step in red.

Tutor, out loud: “Your factorization is correct, but check the signs in your final interval.”

Tutor, in writing:

The student corrects their work.

Student, out loud: “Oh. I tried k = 0 and got minus twelve, and that's not bigger than zero. So the answer is the outside part, not the middle?”

Tutor, out loud: “Yes, the outside intervals are correct. Your revised answer excludes the endpoints, as it should.”

Tutor, in writing:

The tutor confirms the solution is complete.

The problem: Chords AB and CD of a circle intersect at P. AP = 4, PB = 9 and CD = 15. Find CP and PD.

The student writes on the page.

The tutor underlines a step in red.

Tutor, out loud: “Check which two lengths the intersecting-chords theorem multiplies. The full chord is not one of those two segments.”

Tutor, in writing:

The student corrects their work.

Student, out loud: “Oh, so it's the two pieces of CD on either side of P, not the whole chord. Then I need a quadratic.”

Tutor, out loud: “Yes, you corrected the chord relationship, and your quadratic and factoring are right. What does your other root mean for the two labeled lengths?”

Tutor, in writing:

The student corrects their work.

Student, out loud: “Right, P could just as well be closer to D. So CP = 12 and PD = 3 works too.”

Tutor, out loud: “Exactly. Both assignments work, and the given lengths don't distinguish between them.”

Tutor, in writing:

The tutor confirms the solution is complete.

The problem: Evaluate ∫ x·e^(2x) dx.

The student writes on the page.

The tutor underlines a step in red.

Tutor, out loud: “Your integration by parts setup is right. Check the coefficient in the last term by differentiating it.”

Tutor, in writing:

Student, out loud: “Why is the last line wrong? Isn't the integral of e to the 2x just e to the 2x?”

The tutor underlines a step in red.

Tutor, out loud: “Not quite: the chain rule introduces an extra factor when you differentiate the exponential. Your earlier value for the antiderivative already accounts for that; use it to reconsider the last term.”

Tutor, in writing:

The student corrects their work.

Tutor, out loud: “Yes, that correction is right. Your final antiderivative is correct.”

Tutor, in writing:

The tutor confirms the solution is complete.

The problem: A 2.0 kg block slides down a 30° incline. The coefficient of kinetic friction is 0.20. Find the block's acceleration. Use g = 9.8 m/s².

The student writes on the page.

The tutor underlines a step in red.

Tutor, out loud: “Your force directions look good, but check which component of weight is perpendicular to the incline. That affects the friction and acceleration too.”

Tutor, in writing:

The student corrects their work.

Student, out loud: “Right, the normal force only balances the part of gravity that pushes into the ramp. That's the cosine part, so I redid it.”

Tutor, out loud: “Yes, your corrected normal force and friction are right, and your acceleration is correct down the incline.”

Tutor, in writing:

The tutor confirms the solution is complete.

The problem: 10.0 g of hydrogen gas reacts with 64.0 g of oxygen gas. What mass of water forms?

The student writes on the page.

The tutor underlines a step in red.

Tutor, out loud: “Your mole conversions are correct. Check whether both reactants are available in the proportions needed before using all the hydrogen.”

Tutor, in writing:

The student corrects their work.

Student, out loud: “Wait, the oxygen runs out first. 4.96 moles of hydrogen would need 2.48 moles of oxygen, and I only have 2.”

Tutor, out loud: “Exactly. Oxygen is limiting, and your corrected water mass is right.”

Tutor, in writing:

The tutor confirms the solution is complete.