A student finds the roots of a quadratic characteristic equation. Given leading coefficient (a) = 2.8000; linear coefficient (b) = 9.7000; constant term (c) = 0.4000, determine the root (x).

Start with the thinking

• The governing relation printed in this handbook section is Roots of a quadratic equation.
• Everything except x is given, so isolate x 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.
• Finding the roots of a quadratic equation ax^2+bx+c=0 uses the quadratic root formula.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$x = \dfrac{-b + \sqrt{b^2 - 4 a c}}{2 a}$$

2. Step 2 — Rearrange symbolically for x:

   $$x = \dfrac{-b + \sqrt{b^2 - 4 a c}}{2 a}$$

3. Step 3 — List the givens: leading coefficient (a) = 2.8000, linear coefficient (b) = 9.7000, constant term (c) = 0.4000.

4. Step 4 — Substitute the given values:

   $$x = \dfrac{-9.7000 + \sqrt{9.7000^2 - 4 2.8000 0.4000}}{2 2.8000}$$

5. Step 5 — Evaluate:

   $$x = -0.0417$$

6. Step 6 — Check: returning x = -0.0417 to

   $$x = \dfrac{-b + \sqrt{b^2 - 4 a c}}{2 a}$$

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

Why the other options are there

• -0.0835 — kept a factor of two that cancels in the correct rearrangement.
• -0.0209 — dropped that same factor in the other direction.
• -0.0459 — rounded an intermediate value before the final step.

Reference: FE Handbook — Quadratic Equation Roots

An engineer solves for the roots of a quadratic polynomial model. Given leading coefficient (a) = 1.4000; linear coefficient (b) = 7.5000; root (x) = -3.4000, determine the constant term (c).

Start with the thinking

• The governing relation printed in this handbook section is Roots of a quadratic equation.
• Everything except c is given, so isolate c 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.
• Finding the roots of a quadratic equation ax^2+bx+c=0 uses the quadratic root formula.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$x = \dfrac{-b + \sqrt{b^2 - 4 a c}}{2 a}$$

2. Step 2 — Rearrange symbolically for c:

   $$c = -a x^2 - b x$$

3. Step 3 — List the givens: leading coefficient (a) = 1.4000, linear coefficient (b) = 7.5000, root (x) = -3.4000.

4. Step 4 — Substitute the given values:

   $$c = -1.4000 -3.4000^2 - 7.5000 -3.4000$$

5. Step 5 — Evaluate:

   $$c = 9.3160$$

6. Step 6 — Check: returning c = 9.3160 to

   $$x = \dfrac{-b + \sqrt{b^2 - 4 a c}}{2 a}$$

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

Why the other options are there

• 18.6320 — kept a factor of two that cancels in the correct rearrangement.
• 4.6580 — dropped that same factor in the other direction.
• 10.2476 — rounded an intermediate value before the final step.

Reference: FE Handbook — Quadratic Equation Roots

A root of the quadratic equation is back-substituted to check the solution. Given leading coefficient (a) = 1.7000; constant term (c) = 0.0000; root (x) = -9.2500, determine the linear coefficient (b).

Start with the thinking

• The governing relation printed in this handbook section is Roots of a quadratic equation.
• Everything except b is given, so isolate b 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.
• Finding the roots of a quadratic equation ax^2+bx+c=0 uses the quadratic root formula.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$x = \dfrac{-b + \sqrt{b^2 - 4 a c}}{2 a}$$

2. Step 2 — Rearrange symbolically for b:

   $$b = -\dfrac{a x^2 + c}{x}$$

3. Step 3 — List the givens: leading coefficient (a) = 1.7000, constant term (c) = 0.0000, root (x) = -9.2500.

4. Step 4 — Substitute the given values:

   $$b = -\dfrac{1.7000 -9.2500^2 + 0.0000}{-9.2500}$$

5. Step 5 — Evaluate:

   $$b = 15.7250$$

6. Step 6 — Check: returning b = 15.7250 to

   $$x = \dfrac{-b + \sqrt{b^2 - 4 a c}}{2 a}$$

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

Why the other options are there

• 31.4500 — kept a factor of two that cancels in the correct rearrangement.
• 7.8625 — dropped that same factor in the other direction.
• 17.2975 — rounded an intermediate value before the final step.

Reference: FE Handbook — Quadratic Equation Roots

A student finds the roots of a quadratic characteristic equation. Given leading coefficient (a) = 2.7000; linear coefficient (b) = 8.3000; constant term (c) = 0.3000, determine the root (x).

Start with the thinking

• The governing relation printed in this handbook section is Roots of a quadratic equation.
• Everything except x is given, so isolate x 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.
• Finding the roots of a quadratic equation ax^2+bx+c=0 uses the quadratic root formula.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$x = \dfrac{-b + \sqrt{b^2 - 4 a c}}{2 a}$$

2. Step 2 — Rearrange symbolically for x:

   $$x = \dfrac{-b + \sqrt{b^2 - 4 a c}}{2 a}$$

3. Step 3 — List the givens: leading coefficient (a) = 2.7000, linear coefficient (b) = 8.3000, constant term (c) = 0.3000.

4. Step 4 — Substitute the given values:

   $$x = \dfrac{-8.3000 + \sqrt{8.3000^2 - 4 2.7000 0.3000}}{2 2.7000}$$

5. Step 5 — Evaluate:

   $$x = -0.0366$$

6. Step 6 — Check: returning x = -0.0366 to

   $$x = \dfrac{-b + \sqrt{b^2 - 4 a c}}{2 a}$$

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

Why the other options are there

• -0.0732 — kept a factor of two that cancels in the correct rearrangement.
• -0.0183 — dropped that same factor in the other direction.
• -0.0402 — rounded an intermediate value before the final step.

Reference: FE Handbook — Quadratic Equation Roots

An engineer solves for the roots of a quadratic polynomial model. Given leading coefficient (a) = 2.8000; linear coefficient (b) = 6.5000; root (x) = -3.0500, determine the constant term (c).

Start with the thinking

• The governing relation printed in this handbook section is Roots of a quadratic equation.
• Everything except c is given, so isolate c 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.
• Finding the roots of a quadratic equation ax^2+bx+c=0 uses the quadratic root formula.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$x = \dfrac{-b + \sqrt{b^2 - 4 a c}}{2 a}$$

2. Step 2 — Rearrange symbolically for c:

   $$c = -a x^2 - b x$$

3. Step 3 — List the givens: leading coefficient (a) = 2.8000, linear coefficient (b) = 6.5000, root (x) = -3.0500.

4. Step 4 — Substitute the given values:

   $$c = -2.8000 -3.0500^2 - 6.5000 -3.0500$$

5. Step 5 — Evaluate:

   $$c = -6.2220$$

6. Step 6 — Check: returning c = -6.2220 to

   $$x = \dfrac{-b + \sqrt{b^2 - 4 a c}}{2 a}$$

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

Why the other options are there

• -12.4440 — kept a factor of two that cancels in the correct rearrangement.
• -3.1110 — dropped that same factor in the other direction.
• -6.8442 — rounded an intermediate value before the final step.

Reference: FE Handbook — Quadratic Equation Roots

A root of the quadratic equation is back-substituted to check the solution. Given leading coefficient (a) = 2.6000; constant term (c) = 1.4000; root (x) = -9.3500, determine the linear coefficient (b).

Start with the thinking

• The governing relation printed in this handbook section is Roots of a quadratic equation.
• Everything except b is given, so isolate b 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.
• Finding the roots of a quadratic equation ax^2+bx+c=0 uses the quadratic root formula.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$x = \dfrac{-b + \sqrt{b^2 - 4 a c}}{2 a}$$

2. Step 2 — Rearrange symbolically for b:

   $$b = -\dfrac{a x^2 + c}{x}$$

3. Step 3 — List the givens: leading coefficient (a) = 2.6000, constant term (c) = 1.4000, root (x) = -9.3500.

4. Step 4 — Substitute the given values:

   $$b = -\dfrac{2.6000 -9.3500^2 + 1.4000}{-9.3500}$$

5. Step 5 — Evaluate:

   $$b = 24.4597$$

6. Step 6 — Check: returning b = 24.4597 to

   $$x = \dfrac{-b + \sqrt{b^2 - 4 a c}}{2 a}$$

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

Why the other options are there

• 48.9195 — kept a factor of two that cancels in the correct rearrangement.
• 12.2299 — dropped that same factor in the other direction.
• 26.9057 — rounded an intermediate value before the final step.

Reference: FE Handbook — Quadratic Equation Roots

A student finds the roots of a quadratic characteristic equation. Given leading coefficient (a) = 2.2000; linear coefficient (b) = 7.5000; constant term (c) = 0.6000, determine the root (x).

Start with the thinking

• The governing relation printed in this handbook section is Roots of a quadratic equation.
• Everything except x is given, so isolate x 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.
• Finding the roots of a quadratic equation ax^2+bx+c=0 uses the quadratic root formula.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$x = \dfrac{-b + \sqrt{b^2 - 4 a c}}{2 a}$$

2. Step 2 — Rearrange symbolically for x:

   $$x = \dfrac{-b + \sqrt{b^2 - 4 a c}}{2 a}$$

3. Step 3 — List the givens: leading coefficient (a) = 2.2000, linear coefficient (b) = 7.5000, constant term (c) = 0.6000.

4. Step 4 — Substitute the given values:

   $$x = \dfrac{-7.5000 + \sqrt{7.5000^2 - 4 2.2000 0.6000}}{2 2.2000}$$

5. Step 5 — Evaluate:

   $$x = -0.0820$$

6. Step 6 — Check: returning x = -0.0820 to

   $$x = \dfrac{-b + \sqrt{b^2 - 4 a c}}{2 a}$$

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

Why the other options are there

• -0.1639 — kept a factor of two that cancels in the correct rearrangement.
• -0.0410 — dropped that same factor in the other direction.
• -0.0902 — rounded an intermediate value before the final step.

Reference: FE Handbook — Quadratic Equation Roots

An engineer solves for the roots of a quadratic polynomial model. Given leading coefficient (a) = 2.4000; linear coefficient (b) = 8.4000; root (x) = -1.7000, determine the constant term (c).

Start with the thinking

• The governing relation printed in this handbook section is Roots of a quadratic equation.
• Everything except c is given, so isolate c 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.
• Finding the roots of a quadratic equation ax^2+bx+c=0 uses the quadratic root formula.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$x = \dfrac{-b + \sqrt{b^2 - 4 a c}}{2 a}$$

2. Step 2 — Rearrange symbolically for c:

   $$c = -a x^2 - b x$$

3. Step 3 — List the givens: leading coefficient (a) = 2.4000, linear coefficient (b) = 8.4000, root (x) = -1.7000.

4. Step 4 — Substitute the given values:

   $$c = -2.4000 -1.7000^2 - 8.4000 -1.7000$$

5. Step 5 — Evaluate:

   $$c = 7.3440$$

6. Step 6 — Check: returning c = 7.3440 to

   $$x = \dfrac{-b + \sqrt{b^2 - 4 a c}}{2 a}$$

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

Why the other options are there

• 14.6880 — kept a factor of two that cancels in the correct rearrangement.
• 3.6720 — dropped that same factor in the other direction.
• 8.0784 — rounded an intermediate value before the final step.

Reference: FE Handbook — Quadratic Equation Roots

A root of the quadratic equation is back-substituted to check the solution. Given leading coefficient (a) = 1.8000; constant term (c) = 0.5000; root (x) = -4.4000, determine the linear coefficient (b).

Start with the thinking

• The governing relation printed in this handbook section is Roots of a quadratic equation.
• Everything except b is given, so isolate b 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.
• Finding the roots of a quadratic equation ax^2+bx+c=0 uses the quadratic root formula.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$x = \dfrac{-b + \sqrt{b^2 - 4 a c}}{2 a}$$

2. Step 2 — Rearrange symbolically for b:

   $$b = -\dfrac{a x^2 + c}{x}$$

3. Step 3 — List the givens: leading coefficient (a) = 1.8000, constant term (c) = 0.5000, root (x) = -4.4000.

4. Step 4 — Substitute the given values:

   $$b = -\dfrac{1.8000 -4.4000^2 + 0.5000}{-4.4000}$$

5. Step 5 — Evaluate:

   $$b = 8.0336$$

6. Step 6 — Check: returning b = 8.0336 to

   $$x = \dfrac{-b + \sqrt{b^2 - 4 a c}}{2 a}$$

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

Why the other options are there

• 16.0673 — kept a factor of two that cancels in the correct rearrangement.
• 4.0168 — dropped that same factor in the other direction.
• 8.8370 — rounded an intermediate value before the final step.

Reference: FE Handbook — Quadratic Equation Roots

A student finds the roots of a quadratic characteristic equation. Given leading coefficient (a) = 1.2000; linear coefficient (b) = 6.2000; constant term (c) = 0.5000, determine the root (x).

Start with the thinking

• The governing relation printed in this handbook section is Roots of a quadratic equation.
• Everything except x is given, so isolate x 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.
• Finding the roots of a quadratic equation ax^2+bx+c=0 uses the quadratic root formula.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$x = \dfrac{-b + \sqrt{b^2 - 4 a c}}{2 a}$$

2. Step 2 — Rearrange symbolically for x:

   $$x = \dfrac{-b + \sqrt{b^2 - 4 a c}}{2 a}$$

3. Step 3 — List the givens: leading coefficient (a) = 1.2000, linear coefficient (b) = 6.2000, constant term (c) = 0.5000.

4. Step 4 — Substitute the given values:

   $$x = \dfrac{-6.2000 + \sqrt{6.2000^2 - 4 1.2000 0.5000}}{2 1.2000}$$

5. Step 5 — Evaluate:

   $$x = -0.0819$$

6. Step 6 — Check: returning x = -0.0819 to

   $$x = \dfrac{-b + \sqrt{b^2 - 4 a c}}{2 a}$$

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

Why the other options are there

• -0.1639 — kept a factor of two that cancels in the correct rearrangement.
• -0.0410 — dropped that same factor in the other direction.
• -0.0901 — rounded an intermediate value before the final step.

Reference: FE Handbook — Quadratic Equation Roots