A student finds the roots of a quadratic characteristic equation. Given leading coefficient (a) = 1.3000; linear coefficient (b) = 9.0000; 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) = 1.3000, linear coefficient (b) = 9.0000, constant term (c) = 0.3000.

4. Step 4 — Substitute the given values:

   $$x = \dfrac{-9.0000 + \sqrt{9.0000^2 - 4 1.3000 0.3000}}{2 1.3000}$$

5. Step 5 — Evaluate:

   $$x = -0.0335$$

6. Step 6 — Check: returning x = -0.0335 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

• -0.0670 — kept a factor of two that cancels in the correct rearrangement.
• -0.0167 — dropped that same factor in the other direction.
• -0.0368 — rounded an intermediate value before the final step.

Reference: FE Handbook — Quadratic Equation Roots

A student applies Newton's method for root extraction to a nonlinear equation. Given current estimate (xn) = 1.8000; function value f(xn) (fxn) = 7.7000; derivative value f'(xn) (fpxn) = 14.7000, determine the next estimate (xn1).

Start with the thinking

• The governing relation printed in this handbook section is Newton's method for root extraction.
• Everything except xn1 is given, so isolate xn1 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.
• Newton's method for root extraction improves an estimate xn to xn+1 using the function and its derivative.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$x_{n+1} = x_n - \dfrac{f(x_n)}{f'(x_n)}$$

2. Step 2 — Rearrange symbolically for xn1:

   $$xn_{1} = x_n - \dfrac{f(x_n)}{f'(x_n)}$$

3. Step 3 — List the givens: current estimate (xn) = 1.8000, function value f(xn) (fxn) = 7.7000, derivative value f'(xn) (fpxn) = 14.7000.

4. Step 4 — Substitute the given values:

   $$xn_{1} = x_n - \dfrac{f(x_n)}{f'(x_n)}$$

5. Step 5 — Evaluate:

   $$xn_{1} = 1.2762$$

6. Step 6 — Check: returning xn1 = 1.2762 to

   $$x_{n+1} = x_n - \dfrac{f(x_n)}{f'(x_n)}$$

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

Why the other options are there

• 2.5524 — kept a factor of two that cancels in the correct rearrangement.
• 0.6381 — dropped that same factor in the other direction.
• 1.4038 — rounded an intermediate value before the final step.

Reference: FE Handbook — Newton's Method for Root Extraction

An engineer solves for the roots of a quadratic polynomial model. Given leading coefficient (a) = 1.3000; linear coefficient (b) = 6.7000; root (x) = -2.9000, 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) = 1.3000, linear coefficient (b) = 6.7000, root (x) = -2.9000.

4. Step 4 — Substitute the given values:

   $$c = -1.3000 -2.9000^2 - 6.7000 -2.9000$$

5. Step 5 — Evaluate:

   $$c = 8.4970$$

6. Step 6 — Check: returning c = 8.4970 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

• 16.9940 — kept a factor of two that cancels in the correct rearrangement.
• 4.2485 — dropped that same factor in the other direction.
• 9.3467 — rounded an intermediate value before the final step.

Reference: FE Handbook — Quadratic Equation Roots

Newton's method converges toward the root of a polynomial after successive iterations. Given current estimate (xn) = 2.0000; derivative value f'(xn) (fpxn) = 3.2000; next estimate (xn1) = 1.8200, determine the function value f(xn) (fxn).

Start with the thinking

• The governing relation printed in this handbook section is Newton's method for root extraction.
• Everything except fxn is given, so isolate fxn 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.
• Newton's method for root extraction improves an estimate xn to xn+1 using the function and its derivative.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$x_{n+1} = x_n - \dfrac{f(x_n)}{f'(x_n)}$$

2. Step 2 — Rearrange symbolically for fxn:

   $$fxn = (x_n - x_{n+1}) f'(x_n)$$

3. Step 3 — List the givens: current estimate (xn) = 2.0000, derivative value f'(xn) (fpxn) = 3.2000, next estimate (xn1) = 1.8200.

4. Step 4 — Substitute the given values:

   $$fxn = (x_n - x_{n+1}) f'(x_n)$$

5. Step 5 — Evaluate:

   $$fxn = 0.5760$$

6. Step 6 — Check: returning fxn = 0.5760 to

   $$x_{n+1} = x_n - \dfrac{f(x_n)}{f'(x_n)}$$

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

Why the other options are there

• 1.1520 — kept a factor of two that cancels in the correct rearrangement.
• 0.2880 — dropped that same factor in the other direction.
• 0.6336 — rounded an intermediate value before the final step.

Reference: FE Handbook — Newton's Method for Root Extraction

A root of the quadratic equation is back-substituted to check the solution. Given leading coefficient (a) = 1.6000; constant term (c) = 1.9000; root (x) = -3.2000, 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) = 1.6000, constant term (c) = 1.9000, root (x) = -3.2000.

4. Step 4 — Substitute the given values:

   $$b = -\dfrac{1.6000 -3.2000^2 + 1.9000}{-3.2000}$$

5. Step 5 — Evaluate:

   $$b = 5.7138$$

6. Step 6 — Check: returning b = 5.7138 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

• 11.4275 — kept a factor of two that cancels in the correct rearrangement.
• 2.8569 — dropped that same factor in the other direction.
• 6.2851 — rounded an intermediate value before the final step.

Reference: FE Handbook — Quadratic Equation Roots

An engineer performs one iteration of Newton's method to refine a root estimate. Given current estimate (xn) = 4.0000; function value f(xn) (fxn) = 7.7000; next estimate (xn1) = 1.0000, determine the derivative value f'(xn) (fpxn).

Start with the thinking

• The governing relation printed in this handbook section is Newton's method for root extraction.
• Everything except fpxn is given, so isolate fpxn 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.
• Newton's method for root extraction improves an estimate xn to xn+1 using the function and its derivative.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$x_{n+1} = x_n - \dfrac{f(x_n)}{f'(x_n)}$$

2. Step 2 — Rearrange symbolically for fpxn:

   $$fpxn = \dfrac{f(x_n)}{x_n - x_{n+1}}$$

3. Step 3 — List the givens: current estimate (xn) = 4.0000, function value f(xn) (fxn) = 7.7000, next estimate (xn1) = 1.0000.

4. Step 4 — Substitute the given values:

   $$fpxn = \dfrac{f(x_n)}{x_n - x_{n+1}}$$

5. Step 5 — Evaluate:

   $$fpxn = 2.5667$$

6. Step 6 — Check: returning fpxn = 2.5667 to

   $$x_{n+1} = x_n - \dfrac{f(x_n)}{f'(x_n)}$$

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

Why the other options are there

• 5.1333 — kept a factor of two that cancels in the correct rearrangement.
• 1.2833 — dropped that same factor in the other direction.
• 2.8233 — rounded an intermediate value before the final step.

Reference: FE Handbook — Newton's Method for Root Extraction

A student finds the roots of a quadratic characteristic equation. Given leading coefficient (a) = 2.9000; linear coefficient (b) = 5.1000; constant term (c) = 1.2000, 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) = 2.9000, linear coefficient (b) = 5.1000, constant term (c) = 1.2000.

4. Step 4 — Substitute the given values:

   $$x = \dfrac{-5.1000 + \sqrt{5.1000^2 - 4 2.9000 1.2000}}{2 2.9000}$$

5. Step 5 — Evaluate:

   $$x = -0.2798$$

6. Step 6 — Check: returning x = -0.2798 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

• -0.5596 — kept a factor of two that cancels in the correct rearrangement.
• -0.1399 — dropped that same factor in the other direction.
• -0.3078 — rounded an intermediate value before the final step.

Reference: FE Handbook — Quadratic Equation Roots

A student applies Newton's method for root extraction to a nonlinear equation. Given current estimate (xn) = 4.5000; function value f(xn) (fxn) = 14.9000; derivative value f'(xn) (fpxn) = 7.7000, determine the next estimate (xn1).

Start with the thinking

• The governing relation printed in this handbook section is Newton's method for root extraction.
• Everything except xn1 is given, so isolate xn1 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.
• Newton's method for root extraction improves an estimate xn to xn+1 using the function and its derivative.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$x_{n+1} = x_n - \dfrac{f(x_n)}{f'(x_n)}$$

2. Step 2 — Rearrange symbolically for xn1:

   $$xn_{1} = x_n - \dfrac{f(x_n)}{f'(x_n)}$$

3. Step 3 — List the givens: current estimate (xn) = 4.5000, function value f(xn) (fxn) = 14.9000, derivative value f'(xn) (fpxn) = 7.7000.

4. Step 4 — Substitute the given values:

   $$xn_{1} = x_n - \dfrac{f(x_n)}{f'(x_n)}$$

5. Step 5 — Evaluate:

   $$xn_{1} = 2.5649$$

6. Step 6 — Check: returning xn1 = 2.5649 to

   $$x_{n+1} = x_n - \dfrac{f(x_n)}{f'(x_n)}$$

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

Why the other options are there

• 5.1299 — kept a factor of two that cancels in the correct rearrangement.
• 1.2825 — dropped that same factor in the other direction.
• 2.8214 — rounded an intermediate value before the final step.

Reference: FE Handbook — Newton's Method for Root Extraction

An engineer solves for the roots of a quadratic polynomial model. Given leading coefficient (a) = 1.9000; linear coefficient (b) = 7.8000; root (x) = -7.5500, 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) = 1.9000, linear coefficient (b) = 7.8000, root (x) = -7.5500.

4. Step 4 — Substitute the given values:

   $$c = -1.9000 -7.5500^2 - 7.8000 -7.5500$$

5. Step 5 — Evaluate:

   $$c = -49.4147$$

6. Step 6 — Check: returning c = -49.4147 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

• -98.8295 — kept a factor of two that cancels in the correct rearrangement.
• -24.7074 — dropped that same factor in the other direction.
• -54.3562 — rounded an intermediate value before the final step.

Reference: FE Handbook — Quadratic Equation Roots

Newton's method converges toward the root of a polynomial after successive iterations. Given current estimate (xn) = 4.3000; derivative value f'(xn) (fpxn) = 6.6000; next estimate (xn1) = 4.1500, determine the function value f(xn) (fxn).

Start with the thinking

• The governing relation printed in this handbook section is Newton's method for root extraction.
• Everything except fxn is given, so isolate fxn 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.
• Newton's method for root extraction improves an estimate xn to xn+1 using the function and its derivative.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$x_{n+1} = x_n - \dfrac{f(x_n)}{f'(x_n)}$$

2. Step 2 — Rearrange symbolically for fxn:

   $$fxn = (x_n - x_{n+1}) f'(x_n)$$

3. Step 3 — List the givens: current estimate (xn) = 4.3000, derivative value f'(xn) (fpxn) = 6.6000, next estimate (xn1) = 4.1500.

4. Step 4 — Substitute the given values:

   $$fxn = (x_n - x_{n+1}) f'(x_n)$$

5. Step 5 — Evaluate:

   $$fxn = 0.9900$$

6. Step 6 — Check: returning fxn = 0.9900 to

   $$x_{n+1} = x_n - \dfrac{f(x_n)}{f'(x_n)}$$

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

Why the other options are there

• 1.9800 — kept a factor of two that cancels in the correct rearrangement.
• 0.4950 — dropped that same factor in the other direction.
• 1.0890 — rounded an intermediate value before the final step.

Reference: FE Handbook — Newton's Method for Root Extraction