Quadratic Equation Calculator

Quadratic equation

ax² + bx + c = 0
where a, b, c are the coefficients of the quadratic equation and a≠0

a — quadratic coefficient
b — linear coefficient
c — constant coefficient or free term

The Discriminant of the quadratic equation D determines the number of solutions and their type: real or complex conjugate.
The discriminant is calculated by the formula:
$D$ = $b^2-4\mathrm{ac}$

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If ${\displaystyle D}$ = ${\displaystyle 0}$, the equation has a unique solution.
${\displaystyle x}$ = ${\displaystyle -<!--\; -\; -->\frac{\mathrm{b}}{\mathrm{2a}}}$

For example, the equation
${\displaystyle x^2+\mathrm{<mn>2</mn>}x+\mathrm{<mn>1</mn>}}$ = ${\displaystyle 0}$
Coefficient values
${\displaystyle a}$ = ${\displaystyle 1}$
${\displaystyle b}$ = ${\displaystyle 2}$
${\displaystyle c}$ = ${\displaystyle 1}$
Discriminant
${\displaystyle D}$ = ${\displaystyle b^2-4\mathrm{ac}}$
${\displaystyle D}$ = ${\displaystyle {\mathrm{<mn>2</mn>}}^2-4\hspace{0.17em}·\hspace{0.17em}\mathrm{<mn>1</mn>}\hspace{0.17em}·\hspace{0.17em}\mathrm{<mn>1</mn>}}$ = ${\displaystyle 0}$
${\displaystyle x}$ = ${\displaystyle -<!--\; -\; -->\frac{\mathrm{b}}{\mathrm{2a}}}$
${\displaystyle x}$ = ${\displaystyle -<!--\; -\; -->\frac{\mathrm{<mn>2</mn>}}{2\hspace{0.17em}·\hspace{0.17em}\mathrm{<mn>1</mn>}}}$ = ${\displaystyle -1}$
${\displaystyle x}$ = ${\displaystyle -1}$

Plot of quadratic function
${\displaystyle y}$ = ${\displaystyle x^2+2\hspace{0.17em}·\hspace{0.17em}x+1}$

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If $D>0$, the equation has two different solutions.
$x_1$ = ${\displaystyle {\displaystyle \frac{-b-\sqrt{D}}{2a}}}$
$x_2$ = ${\displaystyle {\displaystyle \frac{-b+\sqrt{D}}{2a}}}$

For example, the equation
${\displaystyle \mathrm{<mn>2</mn>}{x}^2+\mathrm{<mn>5</mn>}x+\mathrm{<mn>3</mn>}}$ = ${\displaystyle 0}$
Coefficient values
${\displaystyle a}$ = ${\displaystyle 2}$
${\displaystyle b}$ = ${\displaystyle 5}$
${\displaystyle c}$ = ${\displaystyle 3}$

Discriminant
${\displaystyle D}$ = ${\displaystyle b^2-4\mathrm{ac}}$
${\displaystyle D}$ = ${\displaystyle {\mathrm{<mn>5</mn>}}^2-4\hspace{0.17em}·\hspace{0.17em}\mathrm{<mn>2</mn>}\hspace{0.17em}·\hspace{0.17em}\mathrm{<mn>3</mn>}}$ = ${\displaystyle \mathrm{1\; >\; 0}}$
$x_1$ = ${\displaystyle {\displaystyle \frac{-\mathrm{<mn>5</mn>}-\sqrt{\mathrm{<mn>1</mn>}}}{2\hspace{0.17em}·\hspace{0.17em}\mathrm{<mn>2</mn>}}}}$ = ${\displaystyle -\frac{3}{2}}$
$x_2$ = ${\displaystyle {\displaystyle \frac{-\mathrm{<mn>5</mn>}+\sqrt{\mathrm{<mn>1</mn>}}}{2\hspace{0.17em}·\hspace{0.17em}\mathrm{<mn>2</mn>}}}}$ = ${\displaystyle -1}$
$x_1$ = ${\displaystyle -\frac{3}{2}}$ = ${\displaystyle -1.5}$
$x_2$ = ${\displaystyle -1}$

Plot of quadratic function
${\displaystyle y}$ = ${\displaystyle \mathrm{<mn>2</mn>}{x}^2+\mathrm{<mn>5</mn>}x+\mathrm{<mn>3</mn>}}$

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If $D<0$, the equation has two complex conjugate solutions expressed by the same formula as for the positive discriminant.
$x_1$ = ${\displaystyle {\displaystyle \frac{-b-\sqrt{D}}{2a}}}$
$x_2$ = ${\displaystyle {\displaystyle \frac{-b+\sqrt{D}}{2a}}}$

For example, the equation
${\displaystyle 3x^2-x+7}$ = ${\displaystyle 0}$
Coefficient values
${\displaystyle a}$ = ${\displaystyle 3}$
${\displaystyle b}$ = ${\displaystyle -1}$
${\displaystyle c}$ = ${\displaystyle 7}$
Discriminant
${\displaystyle D}$ = ${\displaystyle b^2-4\mathrm{ac}}$
${\displaystyle D}$ = = ${\displaystyle \mathrm{-83\; <\; 0}}$
$x_1$ = ${\displaystyle {\displaystyle \frac{-b-\sqrt{D}}{2a}}}$
$x_2$ = ${\displaystyle {\displaystyle \frac{-b+\sqrt{D}}{2a}}}$
$x_1$ = = ${\displaystyle \frac{1}{6}-\frac{\sqrt{83}\hspace{0.17em}·\hspace{0.17em}i}{6}}$
$x_2$ = = ${\displaystyle \frac{1}{6}+\frac{\sqrt{83}\hspace{0.17em}·\hspace{0.17em}i}{6}}$

Plot of quadratic function
${\displaystyle y}$ = ${\displaystyle 3x^2-x+7}$

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Incomplete quadratic equation

An incomplete quadratic equation is characterized by the fact that at least one of the coefficients b or c is zero.

The equation ax² + bx = 0
Let's take an example of the equation ${\displaystyle \mathrm{<mn>2</mn>}{x}^2+\mathrm{<mn>6</mn>}x}$ = ${\displaystyle 0}$
Coefficient values
${\displaystyle a}$ = ${\displaystyle 2}$
${\displaystyle b}$ = ${\displaystyle 6}$
An incomplete quadratic equation of the form ax² + bx = 0, where b≠0 has two real solutions:
$x_1$ = ${\displaystyle 0}$ and $x_2$ = ${\displaystyle -<!--\; -\; -->\frac{b}{\mathrm{}a}}$
Solution
$x_1$ = ${\displaystyle 0}$
$x_2$ = ${\displaystyle -<!--\; -\; -->\frac{\mathrm{<mn>6</mn>}}{\mathrm{}\mathrm{<mn>2</mn>}}}$ = ${\displaystyle -3}$

Plot of quadratic function
${\displaystyle y}$ = ${\displaystyle \mathrm{<mn>2</mn>}{x}^2+\mathrm{<mn>6</mn>}x}$

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The equation ax² + c = 0
Let's take an example of the equation ${\displaystyle \frac{x^2}{2}-5}$ = ${\displaystyle 0}$
Coefficient values
${\displaystyle a}$ = ${\displaystyle \frac{1}{2}}$
${\displaystyle c}$ = ${\displaystyle -5}$
An incomplete quadratic equation of the form ax² + c = 0 and has two real solutions if ${\displaystyle \frac{c}{a}<0}$ and two complex solutions if ${\displaystyle \frac{c}{a}>0}$.
${\displaystyle \frac{c}{a}}$ = = ${\displaystyle -10}$
Solution
${\displaystyle \frac{c}{a}<0}$, the equation has two different solutions:
$x_1$ = ${\displaystyle {\displaystyle -\sqrt{-\frac{c}{a}}}}$
$x_2$ = ${\displaystyle {\displaystyle \sqrt{-\frac{c}{a}}}}$
$x_1$ = = ${\displaystyle -\sqrt{10}}$
$x_2$ = = ${\displaystyle \sqrt{10}}$
$x_1$ = ${\displaystyle -\sqrt{10}}$ = ${\displaystyle -3.16227766016838}$
$x_2$ = ${\displaystyle \sqrt{10}}$ = ${\displaystyle 3.16227766016838}$

Plot of quadratic function
${\displaystyle y}$ = ${\displaystyle \frac{x^2}{2}-5}$