Systems of linear equations (up to 3 vars) and polynomial equations via technology practice questions.

Solve the quadratic equation x² − 5x + 6 = 0.

• x = 1 or x = 6
• x = 2 or x = 3
• x = −2 or x = −3
• x = 1 or x = 5

Answer: x = 2 or x = 3

Factorising: (x − 2)(x − 3) = 0, so x = 2 or x = 3 (they sum to 5 and multiply to 6).

A cinema charges $a for adults, $c for children and $s for seniors. On Friday, 3 adult, 2 child and 1 senior tickets earn $57. On Saturday, 1 adult, 4 child and 2 senior tickets earn $54. On Sunday, 2 adult, 1 child and 3 senior tickets earn $47. Using technology, determine the three prices.

• a = 12, c = 8, s = 5
• a = 8, c = 12, s = 5
• a = 12, c = 5, s = 8
• a = 10, c = 9, s = 5

Answer: a = 12, c = 8, s = 5

Solving the system 3a + 2c + s = 57, a + 4c + 2s = 54, 2a + c + 3s = 47 with a GDC gives a = 12, c = 8, s = 5. Check in any equation: 3(12) + 2(8) + 5 = 57. ✓

Using your GDC's polynomial solver, determine which of the following is a solution of 2x³ − 9x² + x + 12 = 0.

• x = −2
• x = 2
• x = 3
• x = 4

Answer: x = 4

The solver returns x = −1, x = 1.5 and x = 4; substituting x = 4 gives 128 − 144 + 4 + 12 = 0. None of the other listed values satisfies the equation.

A shop sells small ($5), medium ($8) and large ($12) gift boxes. In one day it sells 40 boxes in total, sells twice as many small boxes as large boxes, and takes in $302. With s, m and l the numbers of small, medium and large boxes, which system models this situation?

• s + m + l = 40; 2s = l; 5s + 8m + 12l = 302
• s + m + l = 40; s = 2l; 5s + 8m + 12l = 302
• s + m + l = 302; s = 2l; 5s + 8m + 12l = 40
• s + m + l = 40; s = 2l; 12s + 8m + 5l = 302

Answer: s + m + l = 40; s = 2l; 5s + 8m + 12l = 302

"Twice as many small as large" means s = 2l (a classic reversal trap: 2s = l says the opposite), and each price must multiply its own quantity in the revenue equation. Solving gives s = 18, m = 13, l = 9.

State the number of real solutions of the equation x⁴ − 5x² + 4 = 0.

• 4
• 2
• 0
• 3

Answer: 4

Substituting u = x² gives u² − 5u + 4 = 0, so u = 1 or u = 4, giving x = ±1 and x = ±2: four real solutions. A GDC polynomial solver or graph confirms four x-intercepts.

A company's weekly profit, in hundreds of dollars, from selling x units is modelled by P(x) = −2x² + 60x − 250. Using technology, determine the two break-even values of x (where P(x) = 0).

• x = 5 and x = 20
• x = 10 and x = 25
• x = 5 and x = 25
• x = 15 and x = 25

Answer: x = 5 and x = 25

Solving −2x² + 60x − 250 = 0 (equivalently x² − 30x + 125 = 0) with a GDC gives x = 5 and x = 25. Between these values the parabola is above the x-axis, so the company makes a profit.