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Classic CAS manual · Chapter 9 of 16

Solving equations

Solve one equation or a system, inequalities, exact and numeric solutions, and complex solutions.

9.1Solve an equation#

solve(equation, variable) returns every real solution as exact values. Type the equation with =. Find solve in the Algebra menu (3, 1).

ax2+bx+c=0 ⇒ x=−b±b2−4ac2aax^2+bx+c=0\ \Rightarrow\ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}
The quadratic formula, which solve applies for you.
Example 9.1

A quadratic

  1. Type solve(x²−5x+6=0, x).

    solve(x^2▶−5x+6=0,x)enter
Result of the example "A quadratic" The display shows: solve(x^2−5x+6=0,x) / x=2 or x=3.
You should seex=2 or x=3
Example 9.2

Irrational roots stay exact

x2=2⇒x=±2x^{2}=2\Rightarrow x=\pm\sqrt{2}

  1. Type solve(x²=2, x).

    solve(x^2▶=2,x)enter
Result of the example "Irrational roots stay exact" The display shows: solve(x^2=2,x) / x=-√2 or x=√2.
You should seex=-√2 or x=√2
Example 9.3

A cubic

  1. Type solve(x³−6x²+11x−6=0, x).

    solve(x^3▶−6x^2▶+11x−6=0,x)enter
Result of the example "A cubic" The display shows: solve(x^3−6x^2+11x−6=0,x) / x=1 or x=2 or x=3.
You should seex=1 or x=2 or x=3
Example 9.4

Solve for a letter

  1. Type solve(a·x+b=0, x).

    solve(a×x+b=0,x)enter
Result of the example "Solve for a letter" The display shows: solve(a·x+b=0,x) / x=-b/a.
You should seex=-b/a
Example 9.5

Absolute value equation

  1. Type solve(abs(x−1)=3, x).

    solve(abs(x−1)=3,x)enter
Result of the example "Absolute value equation" The display shows: solve(abs(x−1)=3,x) / x=-2 or x=4.
You should seex=-2 or x=4
Example 9.6

Square root equation

  1. Type solve(√(x+2)=x, x).

    solve(ctrl√x²(x+2)▶=x,x)enter
Result of the example "Square root equation" The display shows: solve(√((x+2))=x,x) / x=2.
You should seex=2
Example 9.7

An exponential

ex=5⇒x=ln⁡5e^{x}=5\Rightarrow x=\ln 5

  1. Type solve(e^x = 5, x).

    solve(e^x▶=5,x)enter
Result of the example "An exponential" The display shows: solve(e^x=5,x) / x=ln(5).
You should seex=ln(5)
Example 9.8

No real solution

  1. Type solve(x²+x+1=0, x).

    solve(x^2▶+x+1=0,x)enter
Result of the example "No real solution" The display shows: solve(x^2+x+1=0,x) / false.
You should seefalse

"false" means no real number makes the equation true. Use cSolve for complex answers.

Example 9.9

Always true

  1. Type solve(0=0, x).

    solve(0=0,x)enter
Result of the example "Always true" The display shows: solve(0=0,x) / true.
You should seetrue

9.2Trig equations#

Trig equations give the general solution. The symbol n1 is any whole number.

Example 9.10

Sine equals one half

sin⁡x=12⇒x=π6+2πn or x=5π6+2πn\sin x=\tfrac12\Rightarrow x=\tfrac{\pi}{6}+2\pi n\ \text{or}\ x=\tfrac{5\pi}{6}+2\pi n

  1. Type solve(sin(x)=1/2, x).

    solve(sin(x)=1÷2,x)enter
Result of the example "Sine equals one half" The display shows: solve(sin(x)=1/2,x) / x=2·π·n1+π/6 or x=2·π·n1+5·π/6.
You should seex=2·π·n1+π/6 or x=2·π·n1+5·π/6

9.3Solve an inequality#

Type < or > from the relations menu (ctrl≠≥▸=, then 3 for <, 4 for >).

Example 9.11

A linear inequality

  1. Type solve(2x+1 < 5, x).

    solve(2x+1ctrl≠≥▸=35,x)enter
Result of the example "A linear inequality" The display shows: solve(2x+1<5,x) / x<2.
You should seex<2

9.4Solve a system of equations#

Join the equations with the word and (typed with the letter keys and spaces) and list the unknowns in curly braces (shift{( … shift})).

Example 9.12

Two equations

  1. Type solve(x+y=3 and x−y=1, {x,y}).

    solve(x+y=3␣and␣x−y=1,shift{(x,yshift}))enter
Result of the example "Two equations" The display shows: solve(x+y=3 and x−y=1,{x,y}) / x=2 and y=1.
You should seex=2 and y=1
Example 9.13

Another pair

  1. Type solve(2x+y=5 and x−y=1, {x,y}).

    solve(2x+y=5␣and␣x−y=1,shift{(x,yshift}))enter
Result of the example "Another pair" The display shows: solve(2x+y=5 and x−y=1,{x,y}) / x=2 and y=1.
You should seex=2 and y=1
Example 9.14

Three equations

  1. Type solve(x+y+z=6 and x−y=0 and y+z=5, {x,y,z}).

    solve(x+y+z=6␣and␣x−y=0␣and␣y+z=5,shift{(x,y,zshift}))enter
Result of the example "Three equations" The display shows: solve(x+y+z=6 and x−y=0 and y+z=5,{x,y,z}) / x=1 and y=1 and z=4.
You should seex=1 and y=1 and z=4

9.5Find zeros#

zeros(expression, variable) lists where an expression equals zero.

Example 9.15

Zeros of a quadratic

  1. Type zeros(x²−4, x).

    zeros(x^2▶−4,x)enter
Result of the example "Zeros of a quadratic" The display shows: zeros(x^2−4,x) / {-2,2}.
You should see{-2,2}
Example 9.16

Zeros of a cubic

  1. Type zeros(x³−x, x).

    zeros(x^3▶−x,x)enter
Result of the example "Zeros of a cubic" The display shows: zeros(x^3−x,x) / {-1,0,1}.
You should see{-1,0,1}

9.6Solve numerically#

nSolve(equation, variable, guess) finds one decimal solution near your guess. Use it when exact solving fails.

Example 9.17

Cosine equals x

  1. Type nSolve(cos(x)=x, x, 1).

    nsolve(cos(x)=x,x,1)enter
Result of the example "Cosine equals x" The display shows: nsolve(cos(x)=x,x,1) / x=0.739085.
You should seex=0.739085

9.7Complex solutions#

cSolve, cZeros and cFactor work with complex numbers even in Real mode.

Example 9.18

Complex roots

  1. Type cSolve(x²+1=0, x).

    csolve(x^2▶+1=0,x)enter
Result of the example "Complex roots" The display shows: csolve(x^2+1=0,x) / x=-i or x=i.
You should seex=-i or x=i
Example 9.19

Complex roots with a real part

  1. Type cSolve(x²+2x+5=0, x).

    csolve(x^2▶+2x+5=0,x)enter
Result of the example "Complex roots with a real part" The display shows: csolve(x^2+2x+5=0,x) / x=-1-2·i or x=-1+2·i.
You should seex=-1-2·i or x=-1+2·i
Example 9.20

Cube roots of one

  1. Type cSolve(x³=1, x).

    csolve(x^3▶=1,x)enter
Result of the example "Cube roots of one" The display shows: csolve(x^3=1,x) / x=-1/2-√3·i/2 or x=-1/2+√3·i/2 or x=1.
You should seex=1
Example 9.21

Complex zeros

  1. Type cZeros(x²+4, x).

    czeros(x^2▶+4,x)enter
Result of the example "Complex zeros" The display shows: czeros(x^2+4,x) / {-2·i,2·i}.
You should see{-2·i,2·i}

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