Equation Calculator
With this calculator, you can solve linear, quadratic, or cubic equations online. Examples of calculations can be found in the corresponding section.
Solving Equations
An equation is an equality with a variable (or unknown). An equation with one variable $x$ is generally written as: $f(x) = g(x)$.
A solution (or root) of an equation is a value of the variable that makes the equation a true numerical equality. To solve an equation means to find all its solutions or prove that there are none.
To solve an equation on the calculator: first, enter the part of the equation before the = sign, press the x=y button, enter the remaining part of the equation, and press the = button to perform the calculations. For example, for the equation $2x - 4 = 0$, the root is $x = 2$. Here’s how this result was obtained using the equation calculator:
$$2x - 4 = 0 \\ \Downarrow \\ x = 2$$
2 x - 4 x=y 0 =
Linear Equations
A linear equation with one unknown is an equation of the following form:
$$ax + b = 0,$$
where
- $x$ is the unknown,
- $a$ is the coefficient of the unknown,
- $b$ is the constant term of the equation.
Linear equations are the simplest type of algebraic equations, and their solution involves performing basic arithmetic operations.
Solution examples:
$$4x - 16 = 0 \\ \Downarrow \\ x = 4$$
4 x - 1 6 x=y 0 =
$$3x + 2 = 6 - x \\ \Downarrow \\ x = 1$$
3 x + 2 x=y 6 - x =
Quadratic Equations
A quadratic equation is an equation of the following form:
$$ax^2 + bx + c = 0 \ \ \ (a \ne 0)$$
Solving quadratic equations on the calculator:
$$4x^2 - 6x + 2 = 0 \\ \Downarrow \\ x_1 = 1, \ x_2 = 0.5$$
4 x x2 - 6 x
+ 2 x=y 0 =
$$x^2 - 3x - 4 = 0 \\ \Downarrow \\ x_1 = 4, \ x_2 = -1$$
x x2 - 3 x - 4 x=y 0 =
$$x^2 - 4x + 4 = 0 \\ \Downarrow \\ x_1 = x_2 = x = 2$$
x x2 - 4 x + 4 x=y 0 =
Coefficient Ratios
There are quadratic equations with coefficients in specific ratios that allow for much simpler solutions.
$$a + b + c = 0 \\ \Downarrow \\ x_1 = 1, \ x_2 = \frac {c} {a}$$
$$a + c = b \\ \Downarrow \\ x_1 = -1, \ x_2 = - \frac {c} {a}$$
The roots of such equations can also be found using a regular calculator.
Discriminant
The discriminant is used to find the roots of a quadratic equation. The formula for calculating the discriminant:
$$D = b^2 - 4ac$$
The formula for finding the roots using the discriminant:
$$x_{1,2} = \frac {-b \pm \sqrt D} {2a}$$
If $D > 0$, the equation has two distinct roots. For example:
$$4x^2 - 6x + 2 = 0 \\ \Downarrow \\ a = 4, \ b = -6, \ c = 2 \\ \Downarrow \\ D = (-6)^2 - 4 × 4 × 2 = 4 \\ \Downarrow \\ x_{1,2} = \frac {-(-6) \pm \sqrt 4} {2 × 4} \\ \Downarrow \\ x_1 = \frac {6 + 2} {8} = 1, \ x_2 = \frac {6 - 2} {8} = 0.5$$
If $D = 0$, the equation has one root (or two equal roots). For example:
$$x^2 - 4x + 4 = 0 \\ \Downarrow \\ a = 1, \ b = -4, \ c = 4 \\ \Downarrow \\ D = (-4)^2 - 4 × 1 × 4 = 0 \\ \Downarrow \\ x_{1,2} = \frac {-(-4) \pm \sqrt 0} {2 × 1} \\ \Downarrow \\ x_1 = x_2 = x = \frac {4 \pm 0} {2} = 2$$
If $D < 0$, the equation has no roots in the set of real numbers:
$$x^2 + 4x + 5 = 0 \\ \Downarrow \\ a = 1, \ b = 4, \ c = 5 \\ \Downarrow \\ D = 6^2 - 4 × 1 × 5 = -4 \\ \Downarrow \\ x_{1,2} = \frac {-4 \pm \sqrt {-4}} {2 × 1} \\ \Downarrow \\ x_1 = \frac {-4 + 2i} {2} = -2 + i, \\ x_2 = \frac {-4 - 2i} {2} = -2 - i$$
Vieta’s Theorem
Vieta’s theorem establishes simple algebraic relationships (Vieta’s formulas) between the roots $x_1, x_2$ of a quadratic equation and its coefficients $a, b, c$. Using these formulas, one can find the roots if the coefficients are known or calculate the coefficients if the roots are known.
Vieta’s formulas:
$$\begin {cases} x_1 + x_2 = - \dfrac {b} {a} \\ x_1 x_2 = \dfrac {c} {a} \end {cases}$$
Biquadratic Equations
A biquadratic equation is an equation of the following form:
$$ax^4 + bx^2 + c = 0 \ \ \ (a \ne 0)$$
If we make the substitution $x^2 = y \ (y \ge 0)$, we obtain a quadratic equation for which we can find the roots $y_1, y_2$. The roots of the biquadratic equation are then found as:
$$x_{1,2} = \pm \sqrt {y_1}, \ \ \ x_{3,4} = \pm \sqrt {y_2}$$
Cubic Equations
A cubic equation is an equation of the following form:
$$ax^3 + bx^2 + cx + d = 0 \ \ \ (a \ne 0)$$
Solution example:
$$x^3 - 6x^2 - 31x + 120 = 0 \\ \Downarrow \\ x_1 = -5, x_2 = 3, x_3 = 8$$
x x3 - 6 x x2 - 3 1 x
+ 1 2 0 x=y 0 =
If a cubic equation is divided by $a$ and $x$ is replaced by $y - \frac {b} {3a}$, it takes the following simpler form:
$$y^3 + py + q = 0,$$
where
$y = x + \frac {b} {3a}$,
$p = \frac {c} {a} - \frac {b^2} {3a^2} = \frac {3ac - b^2} {3a^2}$,
$q = \frac {2b^3} {27a^3} - \frac {bc} {3a^2} + \frac {d} {a} = \frac {2b^3 - 9abc + 27a^2d} {27a^3}$.
Cardano’s Formula
If a cubic equation has the following form:
$$y^3 + py + q = 0,$$
then Cardano’s formula can be used to find the roots of this equation:
$$x = \sqrt[3] {- \frac {q} {2} + \sqrt {\frac {q^2} {4} + \frac {p^3} {27}}} + \\ + \ \sqrt[3] {- \frac {q} {2} - \sqrt {\frac {q^2} {4} + \frac {p^3} {27}}}$$