Cubic Equation worksheet

1. Roots: x = 4.170086, x = 2.311108 or x = 0.518806

1. Swap x for t − b/(3a) to get rid of the x² term. That leaves a simpler cubic, t³ + Pt + Q = 0. First find P = c/a − (b/a)² ÷ 3

   $P$$=13-\frac{{\left(\mathrm{-7}\right)}^2}{3}$$=\mathrm{-3.333333}$

2. Then find Q = 2(b/a)³ ÷ 27 − (b/a)(c/a) ÷ 3 + d/a

   $Q$$=\frac{2\times {\left(\mathrm{-7}\right)}^3}{27}-\frac{\mathrm{-7}\times 13}{3}+\left(\mathrm{-5}\right)$$=\mathrm{-0.074074}$

3. Work out the discriminant Δ = −(4P³ + 27Q²). Its sign tells you how many roots are real

   $\Delta$$=\text{−(}4\times {\left(\mathrm{-3.333333}\right)}^3+27\times {\left(\mathrm{-0.074074}\right)}^2\text{)}$$=148$

4. Δ > 0, so there are three real roots. Viète's cosine method finds them: t = m cos(φ/3 − 120°k). First find m = 2√(−P/3)

   $m$$=2\times \sqrt{\frac{3.333333}{3}}$$=2.108185$

5. Next find the angle φ from cos φ = (3Q / 2P)√(−3/P). Use k = 0, 1, 2 for the three roots, then shift back with x = t − b/(3a)

   $\text{cos}\phi$$=\frac{3\times \left(\mathrm{-0.074074}\right)}{2\times \left(\mathrm{-3.333333}\right)}\times \sqrt{\frac{\mathrm{-3}}{\mathrm{-3.333333}}}$$=0.031623$

6. So the roots of the cubic are

   $\text{x = 4.170086, x = 2.311108 or x = 0.518806}$

2. Roots: x = 0.727756 or x = 1.636122 ± 2.634712i

1. Swap x for t − b/(3a) to get rid of the x² term. That leaves a simpler cubic, t³ + Pt + Q = 0. First find P = c/a − (b/a)² ÷ 3

   $P$$=12-\frac{{\left(\mathrm{-4}\right)}^2}{3}$$=6.666667$

2. Then find Q = 2(b/a)³ ÷ 27 − (b/a)(c/a) ÷ 3 + d/a

   $Q$$=\frac{2\times {\left(\mathrm{-4}\right)}^3}{27}-\frac{\mathrm{-4}\times 12}{3}+\left(\mathrm{-7}\right)$$=4.259259$

3. Work out the discriminant Δ = −(4P³ + 27Q²). Its sign tells you how many roots are real

   $\Delta$$=\text{−(}4\times {6.666667}^3+27\times {4.259259}^2\text{)}$$=\mathrm{-1,675}$

4. Δ < 0, so there is one real root. Cardano's formula finds it, and Newton's method polishes it

   $\mathrm{x₁}$$=0.727756$

5. Divide out (x − x₁). The quadratic that's left gives the complex pair

   $x\text{₂,₃}$$=\text{1.636122 ± 2.634712i}$

6. So the roots of the cubic are

   $\text{x = 0.727756 or x = 1.636122 ± 2.634712i}$

3. Roots: x = 1 or x = 0.75 ± 1.391941i

1. Divide every term by a, so the equation starts with a plain x³

   $x^3+\mathrm{-2.5}\times x^2+4\times x+\left(\mathrm{-2.5}\right)$$=0$

2. Swap x for t − b/(3a) to get rid of the x² term. That leaves a simpler cubic, t³ + Pt + Q = 0. First find P = c/a − (b/a)² ÷ 3

   $P$$=4-\frac{{\left(\mathrm{-2.5}\right)}^2}{3}$$=1.916667$

3. Then find Q = 2(b/a)³ ÷ 27 − (b/a)(c/a) ÷ 3 + d/a

   $Q$$=\frac{2\times {\left(\mathrm{-2.5}\right)}^3}{27}-\frac{\mathrm{-2.5}\times 4}{3}+\left(\mathrm{-2.5}\right)$$=\mathrm{-0.324074}$

4. Work out the discriminant Δ = −(4P³ + 27Q²). Its sign tells you how many roots are real

   $\Delta$$=\text{−(}4\times {1.916667}^3+27\times {\left(\mathrm{-0.324074}\right)}^2\text{)}$$=\mathrm{-31}$

5. Δ < 0, so there is one real root. Cardano's formula finds it, and Newton's method polishes it

   $\mathrm{x₁}$$=1$

6. Divide out (x − x₁). The quadratic that's left gives the complex pair

   $x\text{₂,₃}$$=\text{0.75 ± 1.391941i}$

7. So the roots of the cubic are

   $\text{x = 1 or x = 0.75 ± 1.391941i}$

4. Roots: x = 0.535404 or x = 2.732298 ± 2.368286i

1. Swap x for t − b/(3a) to get rid of the x² term. That leaves a simpler cubic, t³ + Pt + Q = 0. First find P = c/a − (b/a)² ÷ 3

   $P$$=16-\frac{{\left(\mathrm{-6}\right)}^2}{3}$$=4$

2. Then find Q = 2(b/a)³ ÷ 27 − (b/a)(c/a) ÷ 3 + d/a

   $Q$$=\frac{2\times {\left(\mathrm{-6}\right)}^3}{27}-\frac{\mathrm{-6}\times 16}{3}+\left(\mathrm{-7}\right)$$=9$

3. Work out the discriminant Δ = −(4P³ + 27Q²). Its sign tells you how many roots are real

   $\Delta$$=\text{−(}4\times 4^3+27\times 9^2\text{)}$$=\mathrm{-2,443}$

4. Δ < 0, so there is one real root. Cardano's formula finds it, and Newton's method polishes it

   $\mathrm{x₁}$$=0.535404$

5. Divide out (x − x₁). The quadratic that's left gives the complex pair

   $x\text{₂,₃}$$=\text{2.732298 ± 2.368286i}$

6. So the roots of the cubic are

   $\text{x = 0.535404 or x = 2.732298 ± 2.368286i}$

5. Roots: x = 0.5 or x = 1.5 ± 1.658312i

1. Divide every term by a, so the equation starts with a plain x³

   $x^3+\mathrm{-3.5}\times x^2+6.5\times x+\left(\mathrm{-2.5}\right)$$=0$

2. Swap x for t − b/(3a) to get rid of the x² term. That leaves a simpler cubic, t³ + Pt + Q = 0. First find P = c/a − (b/a)² ÷ 3

   $P$$=6.5-\frac{{\left(\mathrm{-3.5}\right)}^2}{3}$$=2.416667$

3. Then find Q = 2(b/a)³ ÷ 27 − (b/a)(c/a) ÷ 3 + d/a

   $Q$$=\frac{2\times {\left(\mathrm{-3.5}\right)}^3}{27}-\frac{\mathrm{-3.5}\times 6.5}{3}+\left(\mathrm{-2.5}\right)$$=1.907407$

4. Work out the discriminant Δ = −(4P³ + 27Q²). Its sign tells you how many roots are real

   $\Delta$$=\text{−(}4\times {2.416667}^3+27\times {1.907407}^2\text{)}$$=\mathrm{-154.6875}$

5. Δ < 0, so there is one real root. Cardano's formula finds it, and Newton's method polishes it

   $\mathrm{x₁}$$=0.5$

6. Divide out (x − x₁). The quadratic that's left gives the complex pair

   $x\text{₂,₃}$$=\text{1.5 ± 1.658312i}$

7. So the roots of the cubic are

   $\text{x = 0.5 or x = 1.5 ± 1.658312i}$

6. Roots: x = 5.060647 or x = 0.969676 ± 0.495325i

1. Swap x for t − b/(3a) to get rid of the x² term. That leaves a simpler cubic, t³ + Pt + Q = 0. First find P = c/a − (b/a)² ÷ 3

   $P$$=11-\frac{{\left(\mathrm{-7}\right)}^2}{3}$$=\mathrm{-5.333333}$

2. Then find Q = 2(b/a)³ ÷ 27 − (b/a)(c/a) ÷ 3 + d/a

   $Q$$=\frac{2\times {\left(\mathrm{-7}\right)}^3}{27}-\frac{\mathrm{-7}\times 11}{3}+\left(\mathrm{-6}\right)$$=\mathrm{-5.740741}$

3. Work out the discriminant Δ = −(4P³ + 27Q²). Its sign tells you how many roots are real

   $\Delta$$=\text{−(}4\times {\left(\mathrm{-5.333333}\right)}^3+27\times {\left(\mathrm{-5.740741}\right)}^2\text{)}$$=\mathrm{-283}$

4. Δ < 0, so there is one real root. Cardano's formula finds it, and Newton's method polishes it

   $\mathrm{x₁}$$=5.060647$

5. Divide out (x − x₁). The quadratic that's left gives the complex pair

   $x\text{₂,₃}$$=\text{0.969676 ± 0.495325i}$

6. So the roots of the cubic are

   $\text{x = 5.060647 or x = 0.969676 ± 0.495325i}$

7. Roots: x = 3.893289 or x = 1.053355 ± 0.829704i

1. Swap x for t − b/(3a) to get rid of the x² term. That leaves a simpler cubic, t³ + Pt + Q = 0. First find P = c/a − (b/a)² ÷ 3

   $P$$=10-\frac{{\left(\mathrm{-6}\right)}^2}{3}$$=\mathrm{-2}$

2. Then find Q = 2(b/a)³ ÷ 27 − (b/a)(c/a) ÷ 3 + d/a

   $Q$$=\frac{2\times {\left(\mathrm{-6}\right)}^3}{27}-\frac{\mathrm{-6}\times 10}{3}+\left(\mathrm{-7}\right)$$=\mathrm{-3}$

3. Work out the discriminant Δ = −(4P³ + 27Q²). Its sign tells you how many roots are real

   $\Delta$$=\text{−(}4\times {\left(\mathrm{-2}\right)}^3+27\times {\left(\mathrm{-3}\right)}^2\text{)}$$=\mathrm{-211}$

4. Δ < 0, so there is one real root. Cardano's formula finds it, and Newton's method polishes it

   $\mathrm{x₁}$$=3.893289$

5. Divide out (x − x₁). The quadratic that's left gives the complex pair

   $x\text{₂,₃}$$=\text{1.053355 ± 0.829704i}$

6. So the roots of the cubic are

   $\text{x = 3.893289 or x = 1.053355 ± 0.829704i}$

8. Roots: x = 0.516248 or x = 2.241876 ± 2.158533i

1. Swap x for t − b/(3a) to get rid of the x² term. That leaves a simpler cubic, t³ + Pt + Q = 0. First find P = c/a − (b/a)² ÷ 3

   $P$$=12-\frac{{\left(\mathrm{-5}\right)}^2}{3}$$=3.666667$

2. Then find Q = 2(b/a)³ ÷ 27 − (b/a)(c/a) ÷ 3 + d/a

   $Q$$=\frac{2\times {\left(\mathrm{-5}\right)}^3}{27}-\frac{\mathrm{-5}\times 12}{3}+\left(\mathrm{-5}\right)$$=5.740741$

3. Work out the discriminant Δ = −(4P³ + 27Q²). Its sign tells you how many roots are real

   $\Delta$$=\text{−(}4\times {3.666667}^3+27\times {5.740741}^2\text{)}$$=\mathrm{-1,087}$

4. Δ < 0, so there is one real root. Cardano's formula finds it, and Newton's method polishes it

   $\mathrm{x₁}$$=0.516248$

5. Divide out (x − x₁). The quadratic that's left gives the complex pair

   $x\text{₂,₃}$$=\text{2.241876 ± 2.158533i}$

6. So the roots of the cubic are

   $\text{x = 0.516248 or x = 2.241876 ± 2.158533i}$

9. Roots: x = 1.682328 or x = 0.658836 ± 1.161541i

1. Divide every term by a, so the equation starts with a plain x³

   $x^3+\mathrm{-3}\times x^2+4\times x+\left(\mathrm{-3}\right)$$=0$

2. Swap x for t − b/(3a) to get rid of the x² term. That leaves a simpler cubic, t³ + Pt + Q = 0. First find P = c/a − (b/a)² ÷ 3

   $P$$=4-\frac{{\left(\mathrm{-3}\right)}^2}{3}$$=1$

3. Then find Q = 2(b/a)³ ÷ 27 − (b/a)(c/a) ÷ 3 + d/a

   $Q$$=\frac{2\times {\left(\mathrm{-3}\right)}^3}{27}-\frac{\mathrm{-3}\times 4}{3}+\left(\mathrm{-3}\right)$$=\mathrm{-1}$

4. Work out the discriminant Δ = −(4P³ + 27Q²). Its sign tells you how many roots are real

   $\Delta$$=\text{−(}4\times 1^3+27\times {\left(\mathrm{-1}\right)}^2\text{)}$$=\mathrm{-31}$

5. Δ < 0, so there is one real root. Cardano's formula finds it, and Newton's method polishes it

   $\mathrm{x₁}$$=1.682328$

6. Divide out (x − x₁). The quadratic that's left gives the complex pair

   $x\text{₂,₃}$$=\text{0.658836 ± 1.161541i}$

7. So the roots of the cubic are

   $\text{x = 1.682328 or x = 0.658836 ± 1.161541i}$

10. Roots: x = 0.740079 or x = 2.629961 ± 1.091124i

1. Swap x for t − b/(3a) to get rid of the x² term. That leaves a simpler cubic, t³ + Pt + Q = 0. First find P = c/a − (b/a)² ÷ 3

   $P$$=12-\frac{{\left(\mathrm{-6}\right)}^2}{3}$$=0$

2. Then find Q = 2(b/a)³ ÷ 27 − (b/a)(c/a) ÷ 3 + d/a

   $Q$$=\frac{2\times {\left(\mathrm{-6}\right)}^3}{27}-\frac{\mathrm{-6}\times 12}{3}+\left(\mathrm{-6}\right)$$=2$

3. Work out the discriminant Δ = −(4P³ + 27Q²). Its sign tells you how many roots are real

   $\Delta$$=\text{−(}4\times 0^3+27\times 2^2\text{)}$$=\mathrm{-108}$

4. Δ < 0, so there is one real root. Cardano's formula finds it, and Newton's method polishes it

   $\mathrm{x₁}$$=0.740079$

5. Divide out (x − x₁). The quadratic that's left gives the complex pair

   $x\text{₂,₃}$$=\text{2.629961 ± 1.091124i}$

6. So the roots of the cubic are

   $\text{x = 0.740079 or x = 2.629961 ± 1.091124i}$