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## The calculator solves any type of quadratic equation, including incomplete quadratic equations, calculates real and complex solutions, and also plots and finds the points of intersection of the parabola with the x-axis.

Equation type
Specify the coefficients a, b, c of the quadratic equation ax² + bx + c = 0, where a≠0.
x² + x + = 0

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

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:
$D$ = ${b}^{2}-4\mathrm{ac}$

If $D$ = $0$, the equation has a unique solution.
$x$ = $-\frac{b}{2a}$

For example, the equation
${x}^{2}+2x+1$ = $0$
Coefficient values
$a$ = $1$
$b$ = $2$
$c$ = $1$
Discriminant
$D$ = ${b}^{2}-4\mathrm{ac}$
$D$ = ${2}^{2}-4 · 1 · 1$ = $0$
$x$ = $-\frac{b}{2a}$
$x$ = $-\frac{2}{2 · 1}$ = $-1$
$x$ = $-1$

$y$ = ${x}^{2}+2 · x+1$
The solution to the quadratic equation is the intersection of the parabola with the x-axis. The equation has one solution, then the parabola on the plot intersects the x-axis at only one point.

If $D>0$, the equation has two different solutions.
${x}_{1}$ = $\frac{-b-\sqrt{D}}{2a}$
${x}_{2}$ = $\frac{-b+\sqrt{D}}{2a}$

For example, the equation
$2{x}^{2}+5x+3$ = $0$
Coefficient values
$a$ = $2$
$b$ = $5$
$c$ = $3$

Discriminant
$D$ = ${b}^{2}-4\mathrm{ac}$
$D$ = ${5}^{2}-4 · 2 · 3$ = $1 > 0$
${x}_{1}$ = $\frac{-5-\sqrt{1}}{2 · 2}$ = $-\frac{3}{2}$
${x}_{2}$ = $\frac{-5+\sqrt{1}}{2 · 2}$ = $-1$
${x}_{1}$ = $-\frac{3}{2}$ = $-1.5$
${x}_{2}$ = $-1$

$y$ = $2{x}^{2}+5x+3$
The solution to the quadratic equation is the intersection of the parabola with the x-axis. The equation has two solutions, so the parabola on the graph intersects the x-axis at two points.

If $D<0$, the equation has two complex conjugate solutions expressed by the same formula as for the positive discriminant.
${x}_{1}$ = $\frac{-b-\sqrt{D}}{2a}$
${x}_{2}$ = $\frac{-b+\sqrt{D}}{2a}$

For example, the equation
$3{x}^{2}-x+7$ = $0$
Coefficient values
$a$ = $3$
$b$ = $-1$
$c$ = $7$
Discriminant
$D$ = ${b}^{2}-4\mathrm{ac}$
$D$ = ${\mathrm{\left(-1\right)}}^{2}-4 · 3 · 7$ = $-83 < 0$
${x}_{1}$ = $\frac{-b-\sqrt{D}}{2a}$
${x}_{2}$ = $\frac{-b+\sqrt{D}}{2a}$
${x}_{1}$ = $\frac{-\mathrm{\left(-1\right)}-\sqrt{\mathrm{\left(-83\right)}}}{2 · 3}$ = $\frac{1}{6}-\frac{\sqrt{83} · i}{6}$
${x}_{2}$ = $\frac{-\mathrm{\left(-1\right)}+\sqrt{\mathrm{\left(-83\right)}}}{2 · 3}$ = $\frac{1}{6}+\frac{\sqrt{83} · i}{6}$

$y$ = $3{x}^{2}-x+7$
Pay attention to the parabola of the graph, it does not intersect the x-axis, therefore, the equation has no real solutions.

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 $2{x}^{2}+6x$ = $0$
Coefficient values
$a$ = $2$
$b$ = $6$
An incomplete quadratic equation of the form ax² + bx = 0, where b≠0 has two real solutions:
${x}_{1}$ = $0$ and ${x}_{2}$ = $-\frac{b}{a}$
Solution
${x}_{1}$ = $0$
${x}_{2}$ = $-\frac{6}{2}$ = $-3$

$y$ = $2{x}^{2}+6x$

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

$y$ = $\frac{{x}^{2}}{2}-5$