MENUEquation and Inequality Solver with Steps
Type an equation or an inequality in one variable. The solver finds the solution and shows every step, the way it is done at school: linear, quadratic, polynomial, rational, absolute value, radical, exponential, logarithmic and basic trigonometric equations, and linear, quadratic, polynomial, rational, absolute value and double inequalities. A formula with more letters is solved for the letter you name: 2x + 3y = 6 for y.
Examples: $3(x - 2) + 4 = 2x - 1$ $x^2 - 5x + 6 = 0$ $2x^2 + 3x - 2 = 0$ $\frac{x+1}{x-2} = 3$ $x^4 - 5x^2 + 4 = 0$ $5 - 3x \le 11$ $x^2 - 5x + 6 < 0$ $\frac{x-1}{x+2} \ge 0$ $1 < 2x + 3 \le 7$ $|2x - 3| = 5$ $|x - 1| < 3$ $\sqrt{x + 3} = x - 3$ $2^{x + 1} = 8$ $4^x - 3 \cdot 2^x + 2 = 0$ $\log_2(x - 1) = 3$ $\log x + \log(x - 3) = 1$ $2\sin x - 1 = 0$ $2x + 3y = 6$ for $y$ $e^x = x + 2$
Solution
Step-by-step solution
How to enter a problem
=makes an equation;<,>,<=and>=(or ≤ and ≥) make an inequality. A double inequality such as1 < 2x + 3 <= 7is solved too.*is multiplication; it can be left out:2x,3(x - 2),(x + 1)(x - 2).x^2is $x^2$.(x+1)/(x-2)is $\frac{x+1}{x-2}$: put the numerator and the denominator in brackets.- Decimals are written with a point:
0.5x + 1.2 = 3. - The variable is $x$, or the one letter of the problem:
2t + 1 = 5is solved for $t$. |x - 1|is the absolute value $|x - 1|$.sqrt(x + 3)is $\sqrt{x + 3}$ andcbrt(x)is $\sqrt[3]{x}$.2^(x + 1)is $2^{x + 1}$: put the exponent in brackets.e^xis $e^x$.log(x)is the logarithm to base 10,ln(x)the natural logarithm, andlog_2(x)the logarithm to base 2.sin(x),cos(2x),tan(x): the angles are in radians, and the answer gives every solution with a whole number $k$.- To solve a formula for one of its letters, add
forand the letter:v = u + a t for t.
Linear equations
Clear the fractions by multiplying both sides by the common denominator, and expand the brackets. Then move the terms with $x$ to one side and the numbers to the other; a term changes its sign when it moves across the equals sign. Collect like terms and divide by the coefficient of $x$:
$3(x - 2) + 4 = 2x - 1 \quad\Rightarrow\quad 3x - 2 = 2x - 1 \quad\Rightarrow\quad 3x - 2x = -1 + 2 \quad\Rightarrow\quad x = 1$
More: Solving linear equations.
Quadratic equations
A quadratic equation $ax^2 + bx + c = 0$ is solved with the discriminant $D = b^2 - 4ac$:
$x_{1,2} = \frac{-b \pm \sqrt{D}}{2a}$
If $D > 0$, there are two roots; if $D = 0$, one root; if $D < 0$, no real roots. Two cases are quicker: when $c = 0$, take $x$ out of the brackets, $x(ax + b) = 0$; when $b = 0$, find $x^2 = -\frac{c}{a}$ and take the square root. When $a = 1$ and the roots are whole numbers, look for two numbers whose product is $c$ and whose sum is $b$: $x^2 - 5x + 6 = (x - 2)(x - 3)$.
More: Quadratic equations.
Polynomial equations
A polynomial equation of a higher degree is solved by factoring: a product is zero when one of its factors is zero. Look for whole roots among the divisors of the constant term. A biquadratic equation $ax^4 + bx^2 + c = 0$ becomes a quadratic one with $t = x^2$.
More: Biquadratic equations, Cubic and quartic equations.
Rational equations
First find the values that make a denominator zero: they cannot be solutions. Then multiply both sides by the common denominator and solve the equation without fractions. At the end, drop every root that makes a denominator zero.
Inequalities
A linear inequality is solved like an equation, with one rule: multiplying or dividing both sides by a negative number reverses the inequality sign. $-3x \le 6 \;\Rightarrow\; x \ge -2$.
Other inequalities are solved with a sign chart. Move all terms to one side and write them as one fraction; do not multiply by an expression with $x$, because its sign is not known. Factor the numerator and the denominator. The expression can change its sign only where a factor is zero, so find the sign between those points and choose the intervals where the inequality holds. The zeros of the denominator never belong to the answer.
More: Linear inequalities, Quadratic inequalities, Polynomial and rational inequalities.
Absolute value equations
Get the absolute value alone on one side. An absolute value is never negative, so $|A| = -3$ has no solution. Otherwise the expression inside equals the number or its opposite: $|2x - 3| = 5$ means $2x - 3 = 5$ or $2x - 3 = -5$, so $x = 4$ or $x = -1$. An inequality $|A| < k$ means $-k < A < k$, and $|A| > k$ means $A < -k$ or $A > k$. When the other side has $x$, as in $|x - 3| = 2x$, check each root: the right side must not be negative.
More: Absolute value equations.
Radical equations
Get the root alone on one side and square both sides, then solve the equation without the root. Squaring can bring in roots that do not solve the equation, so put each root back into the equation: $\sqrt{x + 3} = x - 3$ gives $x + 3 = x^2 - 6x + 9$, with the roots $1$ and $6$, but only $x = 6$ solves the equation. With two roots, square twice.
More: Radicals.
Exponential equations
Write both sides as powers of the same number; then the exponents are equal: $2^{x + 1} = 8 = 2^3$ gives $x + 1 = 3$. When the other side is not a power of the base, take the logarithm: $3^x = 5$ gives $x = \log_3 5$. An equation such as $4^x - 3 \cdot 2^x + 2 = 0$ becomes a quadratic one with $t = 2^x$; a power of a positive number is always positive, so only the positive values of $t$ count.
More: Exponents.
Logarithmic equations
A logarithm is defined only for positive numbers. Write the logarithms as one logarithm, with $\log a + \log b = \log(ab)$, then use the definition: $\log_b A = c$ means $A = b^c$. Solve, and keep only the roots for which every number inside a logarithm is positive: $\log x + \log(x - 3) = 1$ gives $x(x - 3) = 10$, with the roots $5$ and $-2$, and only $x = 5$ is a solution.
More: Logarithms.
Trigonometric equations
Get the sine, the cosine or the tangent alone on one side. The sine and the cosine take values from $-1$ to $1$ only. A value repeats with the period, so there are infinitely many solutions: $\sin x = \frac{1}{2}$ gives $x = \frac{\pi}{6} + 2\pi k$ or $x = \frac{5\pi}{6} + 2\pi k$, where $k$ is any whole number; $\cos x = a$ gives $x = \pm\arccos a + 2\pi k$, and $\tan x = a$ gives $x = \arctan a + \pi k$.
More: Trigonometric functions.
Solving a formula for a letter
Treat the other letters as numbers. Move the terms with the letter to one side and the rest to the other, take the letter out of the brackets and divide by what is left beside it: $2x + 3y = 6$ gives $3y = 6 - 2x$, so $y = \frac{6 - 2x}{3}$. Dividing by an expression with letters needs it not to be zero.
Equations solved numerically
When the unknown is both inside a function or an exponent and outside it, as in $e^x = x + 2$, $\cos x = x$ or $x^x = 4$, no method of algebra gives the roots. They are found numerically: move everything to one side, $f(x) = e^x - x - 2$, and look where $f$ changes its sign; each change of sign holds a root. Newton's method then makes it more and more precise, $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$: from $x_0 = 1.5$ it gives $1.218$, $1.1498$, $1.14620$ and $1.146193$. The other root of $e^x = x + 2$ is $x \approx -1.84141$.
