Depth y in a channel balance satisfies 3y² − 11y + 6 = 0. Which positive root is physically meaningful if the channel is 1.5 m deep?

Start with the thinking

• Both algebraic roots are real; physics rejects one.
• Compute the discriminant before the roots.

Step-by-step solution

1. Quadratic formula

   $$y = [-b \pm \sqrt(b^{2} - 4ac)]/(2a)$$

2. Discriminant

   $$b^{2} - 4ac = 121 - 72 = 49$$

3. Root term

   $$\sqrt49 = 7$$

4. Roots

   $$y = (11 \pm 7)/6, so y_{1} = 3.00 m and y_{2} = 0.667 m$$

5. Physical check

   $$y_{1} = 3.00 m exceeds the 1.5 m channel, so y = 0.667 m$$

Why the other options are there

• 3.00 m (geometry limit ignored)
• 1.83 m (average of the roots)

Reference: FE Reference Handbook — Mathematics — Algebra, quadratic equation

A mathematics problem uses Parabola vertex form. Given coefficient (a) = 2.0000; x-value (x) = 3.5000; vertex x (h) = -2.0000; vertex y (k) = 1.0000, determine the y-value (y).

Start with the thinking

• The governing relation printed in this handbook section is Parabola vertex form.
• Everything except y is given, so isolate y symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Mathematics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = a (x - h)^2 + k$$

2. Step 2 — Rearrange the relation so that y stands alone on the left-hand side.

3. Step 3

   $$List the givens: coefficient (a) = 2.0000, x-value (x) = 3.5000, vertex x (h) = -2.0000, vertex y (k) = 1.0000$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$y = 61.5000$$

6. Step 6 — Check: returning y = 61.5000 to

   $$y = a (x - h)^2 + k$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 123.0 — kept a factor of two that cancels in the correct rearrangement.
• 30.7500 — dropped that same factor in the other direction.
• 67.6500 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → Quadratic Equation

A mathematics problem uses Parabola vertex form. Given x-value (x) = -4.5000; vertex x (h) = 1.0000; vertex y (k) = -0.5000; y-value (y) = 43.2000, determine the coefficient (a).

Start with the thinking

• The governing relation printed in this handbook section is Parabola vertex form.
• Everything except a is given, so isolate a symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Mathematics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = a (x - h)^2 + k$$

2. Step 2 — Rearrange the relation so that a stands alone on the left-hand side.

3. Step 3

   $$List the givens: x-value (x) = -4.5000, vertex x (h) = 1.0000, vertex y (k) = -0.5000, y-value (y) = 43.2000$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$a = 1.4446$$

6. Step 6 — Check: returning a = 1.4446 to

   $$y = a (x - h)^2 + k$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 2.8893 — kept a factor of two that cancels in the correct rearrangement.
• 0.7223 — dropped that same factor in the other direction.
• 1.5891 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → Quadratic Equation

A mathematics problem uses Parabola vertex form. Given coefficient (a) = 1.0000; x-value (x) = -2.5000; vertex x (h) = -3.0000; y-value (y) = 22.1000, determine the vertex y (k).

Start with the thinking

• The governing relation printed in this handbook section is Parabola vertex form.
• Everything except k is given, so isolate k symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Mathematics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = a (x - h)^2 + k$$

2. Step 2 — Rearrange the relation so that k stands alone on the left-hand side.

3. Step 3 — List the givens: coefficient (a) = 1.0000, x-value (x) = -2.5000, vertex x (h) = -3.0000, y-value (y) = 22.1000.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$k = 21.8500$$

6. Step 6 — Check: returning k = 21.8500 to

   $$y = a (x - h)^2 + k$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 43.7000 — kept a factor of two that cancels in the correct rearrangement.
• 10.9250 — dropped that same factor in the other direction.
• 24.0350 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → Quadratic Equation

A mathematics problem uses Parabola vertex form. Given coefficient (a) = 1.9000; x-value (x) = 1.5000; vertex x (h) = -1.0000; vertex y (k) = 1.5000, determine the y-value (y).

Start with the thinking

• The governing relation printed in this handbook section is Parabola vertex form.
• Everything except y is given, so isolate y symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Mathematics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = a (x - h)^2 + k$$

2. Step 2 — Rearrange the relation so that y stands alone on the left-hand side.

3. Step 3

   $$List the givens: coefficient (a) = 1.9000, x-value (x) = 1.5000, vertex x (h) = -1.0000, vertex y (k) = 1.5000$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$y = 13.3750$$

6. Step 6 — Check: returning y = 13.3750 to

   $$y = a (x - h)^2 + k$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 26.7500 — kept a factor of two that cancels in the correct rearrangement.
• 6.6875 — dropped that same factor in the other direction.
• 14.7125 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → Quadratic Equation

A mathematics problem uses Parabola vertex form. Given x-value (x) = 3.5000; vertex x (h) = -1.5000; vertex y (k) = 2.0000; y-value (y) = -41.1000, determine the coefficient (a).

Start with the thinking

• The governing relation printed in this handbook section is Parabola vertex form.
• Everything except a is given, so isolate a symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Mathematics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = a (x - h)^2 + k$$

2. Step 2 — Rearrange the relation so that a stands alone on the left-hand side.

3. Step 3

   $$List the givens: x-value (x) = 3.5000, vertex x (h) = -1.5000, vertex y (k) = 2.0000, y-value (y) = -41.1000$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$a = -1.7240$$

6. Step 6 — Check: returning a = -1.7240 to

   $$y = a (x - h)^2 + k$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• -3.4480 — kept a factor of two that cancels in the correct rearrangement.
• -0.8620 — dropped that same factor in the other direction.
• -1.8964 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → Quadratic Equation

A mathematics problem uses Parabola vertex form. Given coefficient (a) = 1.9000; x-value (x) = -3.5000; vertex x (h) = 2.5000; y-value (y) = 13.5000, determine the vertex y (k).

Start with the thinking

• The governing relation printed in this handbook section is Parabola vertex form.
• Everything except k is given, so isolate k symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Mathematics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = a (x - h)^2 + k$$

2. Step 2 — Rearrange the relation so that k stands alone on the left-hand side.

3. Step 3

   $$List the givens: coefficient (a) = 1.9000, x-value (x) = -3.5000, vertex x (h) = 2.5000, y-value (y) = 13.5000$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$k = -54.9000$$

6. Step 6 — Check: returning k = -54.9000 to

   $$y = a (x - h)^2 + k$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• -109.8 — kept a factor of two that cancels in the correct rearrangement.
• -27.4500 — dropped that same factor in the other direction.
• -60.3900 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → Quadratic Equation

A mathematics problem uses Parabola vertex form. Given coefficient (a) = 1.0000; x-value (x) = 1.0000; vertex x (h) = 2.5000; vertex y (k) = 3.0000, determine the y-value (y).

Start with the thinking

• The governing relation printed in this handbook section is Parabola vertex form.
• Everything except y is given, so isolate y symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Mathematics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = a (x - h)^2 + k$$

2. Step 2 — Rearrange the relation so that y stands alone on the left-hand side.

3. Step 3

   $$List the givens: coefficient (a) = 1.0000, x-value (x) = 1.0000, vertex x (h) = 2.5000, vertex y (k) = 3.0000$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$y = 5.2500$$

6. Step 6 — Check: returning y = 5.2500 to

   $$y = a (x - h)^2 + k$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 10.5000 — kept a factor of two that cancels in the correct rearrangement.
• 2.6250 — dropped that same factor in the other direction.
• 5.7750 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → Quadratic Equation

A mathematics problem uses Parabola vertex form. Given x-value (x) = -4.0000; vertex x (h) = 1.5000; vertex y (k) = -2.5000; y-value (y) = -23.0000, determine the coefficient (a).

Start with the thinking

• The governing relation printed in this handbook section is Parabola vertex form.
• Everything except a is given, so isolate a symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Mathematics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = a (x - h)^2 + k$$

2. Step 2 — Rearrange the relation so that a stands alone on the left-hand side.

3. Step 3

   $$List the givens: x-value (x) = -4.0000, vertex x (h) = 1.5000, vertex y (k) = -2.5000, y-value (y) = -23.0000$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$a = -0.6777$$

6. Step 6 — Check: returning a = -0.6777 to

   $$y = a (x - h)^2 + k$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• -1.3554 — kept a factor of two that cancels in the correct rearrangement.
• -0.3388 — dropped that same factor in the other direction.
• -0.7455 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → Quadratic Equation

A mathematics problem uses Parabola vertex form. Given coefficient (a) = 0.5000; x-value (x) = -4.5000; vertex x (h) = -1.0000; y-value (y) = -30.6000, determine the vertex y (k).

Start with the thinking

• The governing relation printed in this handbook section is Parabola vertex form.
• Everything except k is given, so isolate k symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Mathematics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = a (x - h)^2 + k$$

2. Step 2 — Rearrange the relation so that k stands alone on the left-hand side.

3. Step 3 — List the givens: coefficient (a) = 0.5000, x-value (x) = -4.5000, vertex x (h) = -1.0000, y-value (y) = -30.6000.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$k = -36.7250$$

6. Step 6 — Check: returning k = -36.7250 to

   $$y = a (x - h)^2 + k$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• -73.4500 — kept a factor of two that cancels in the correct rearrangement.
• -18.3625 — dropped that same factor in the other direction.
• -40.3975 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → Quadratic Equation