Quadratic Equations Solver

Coefficients

Input coefficients corresponding to a quadratic function where $aa$, $bb$ and $cc$ are real numbers. $a\mathrm{\ne }0a\; \backslash neq\; 0$

Checkboxes

✔️ Roots

Calculate the roots of $f(x)f(x)$, given that $\mathrm{\Delta }=b^2-4ac\backslash Delta\; =\; b^2\; -\; 4ac$

• If Δ > 0, the function admits two real roots $x_{1}x\_1$ and $x_{2}x\_2$ given by ${\displaystyle \frac{-b\pm \sqrt{b^2-4ac}}{2a}}\backslash dfrac\{-b\; \backslash pm\; \backslash sqrt\{b^2\; -\; 4ac\}\}\{2a\}$
• If Δ = 0, the function admits a single real root $x_{0}x\_0$ given by $-{\displaystyle \frac{b}{2a}}-\backslash dfrac\{b\}\{2a\}$
• If Δ < 0, the function admits two complex roots $x_{1}x\_1$ and $x_{2}x\_2$ given by $\dfrac{-b \pm i \sqrt{-(b^2-4ac)}}{2a}$

✔️ Plot

A graphical representation of $f(x)f(x)$ will be shown according to the x-axis range prescribed in the slider.

If the slider isn't configured, the table will be output with default $xx$ values ranging from -5 to 5.

✔️ Table

A table of $xx$ and $yy$ values will be displayed according to the x-axis range prescribed in the slider.

If the slider isn't configured, the table will be output with default $xx$ values ranging from -5 to 5.

✔️ Vertex

The vertex and the vertex form of the function will be presented:

The vertex of a quadratic equation is $(\alpha ,\beta )(\backslash alpha,\; \backslash beta)$ where $\alpha =-{\displaystyle \frac{b}{2a}}\backslash alpha\; =\; -\backslash dfrac\{b\}\{2a\}$ and $\beta =f(\alpha )\backslash beta\; =\; f(\backslash alpha)$

The vertex form of $f(x)f(x)$ is given by $a(x-\alpha {)}^2+\beta a(x\; -\; \backslash alpha)^2\; +\; \backslash beta$

✔️ Factorisation

The factored form of $a{x}^2+bx+cax^2\; +\; bx\; +\; c$ will be exposed:

• If Δ > 0, $f(x)=a(x-x_1)(x-x_2)f(x)\; =\; a(x\; -\; x\_1)(x\; -\; x\_2)$ where $x_{1}x\_1$ and $x_{2}x\_2$ are the solutions to the equation $a{x}^2+bx+c=0ax^2\; +\; bx\; +\; c\; =\; 0$
• If Δ = 0, $f(x)=a(x-x_0{)}^{2}f(x)\; =\; a(x\; -\; x\_0)^2$ where $x_{0}x\_0$ is the unique solution to the equation $a{x}^2+bx+c=0ax^2\; +\; bx\; +\; c\; =\; 0$
• If Δ < 0, the factorisation is impossible since there are no real solutions to the equation $a{x}^2+bx+c=0ax^2\; +\; bx\; +\; c\; =\; 0$

Simple / Detailed

If you'd like quick and straight-forward results, select Simple

If you prefer full calculations and reasoning behind the results, select Detailed