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:
$D$ = ${b}^{2}-4\mathrm{ac}$
If
$D$ =
$0$,
the equation has a unique solution.
$x$ =
$-\frac{\mathrm{b}}{\mathrm{2a}}$
For example, the equation
${x}^{2}+2x+1$ =
$0$Coefficient values
$a$ =
$1$$b$ =
$2$$c$ =
$1$Discriminant
$D$ =
$b}^{2}-4\mathrm{ac$$D$ =
${2}^{2}-4\hspace{0.17em}\xb7\hspace{0.17em}1\hspace{0.17em}\xb7\hspace{0.17em}1$ =
$0$$x$ =
$-\frac{\mathrm{b}}{\mathrm{2a}}$$x$ =
$-\frac{2}{2\hspace{0.17em}\xb7\hspace{0.17em}1}$ =
$-1$$x$ =
$-1$
Plot of quadratic function$y$ =
${x}^{2}+2\hspace{0.17em}\xb7\hspace{0.17em}x+1$
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.
${x}_{1}$ =
$\displaystyle \frac{-b-\sqrt{D}}{2a}$${x}_{2}$ =
$\displaystyle \frac{-b+\sqrt{D}}{2a}$
For example, the equation
$2{x}^{2}+5x+3$ =
$0$Coefficient values
$a$ =
$2$$b$ =
$5$$c$ =
$3$Discriminant
$D$ =
$b}^{2}-4\mathrm{ac$$D$ =
${5}^{2}-4\hspace{0.17em}\xb7\hspace{0.17em}2\hspace{0.17em}\xb7\hspace{0.17em}3$ =
$\mathrm{1\; >\; 0}$${x}_{1}$ =
$\displaystyle \frac{-5-\sqrt{1}}{2\hspace{0.17em}\xb7\hspace{0.17em}2}$ =
$-\frac{3}{2}$${x}_{2}$ =
$\displaystyle \frac{-5+\sqrt{1}}{2\hspace{0.17em}\xb7\hspace{0.17em}2}$ =
$-1$${x}_{1}$ =
$-\frac{3}{2}$ =
$-1.5$${x}_{2}$ =
$-1$
Plot of quadratic function$y$ =
$2{x}^{2}+5x+3$
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.
${x}_{1}$ =
$\displaystyle \frac{-b-\sqrt{D}}{2a}$${x}_{2}$ =
$\displaystyle \frac{-b+\sqrt{D}}{2a}$
For example, the equation
$3{x}^{2}-x+7$ =
$0$Coefficient values
$a$ =
$3$$b$ =
$-1$$c$ =
$7$Discriminant
$D$ =
$b}^{2}-4\mathrm{ac$$D$ =
${\mathrm{\left(-1\right)}}^{2}-4\hspace{0.17em}\xb7\hspace{0.17em}3\hspace{0.17em}\xb7\hspace{0.17em}7$ =
$\mathrm{-83\; <\; 0}$${x}_{1}$ =
$\displaystyle \frac{-b-\sqrt{D}}{2a}$${x}_{2}$ =
$\displaystyle \frac{-b+\sqrt{D}}{2a}$${x}_{1}$ =
$\displaystyle \frac{-\mathrm{\left(-1\right)}-\sqrt{\mathrm{\left(-83\right)}}}{2\hspace{0.17em}\xb7\hspace{0.17em}3}$ =
$\frac{1}{6}-\frac{\sqrt{83}\hspace{0.17em}\xb7\hspace{0.17em}i}{6}$${x}_{2}$ =
$\displaystyle \frac{-\mathrm{\left(-1\right)}+\sqrt{\mathrm{\left(-83\right)}}}{2\hspace{0.17em}\xb7\hspace{0.17em}3}$ =
$\frac{1}{6}+\frac{\sqrt{83}\hspace{0.17em}\xb7\hspace{0.17em}i}{6}$
Plot of quadratic function$y$ =
$3{x}^{2}-x+7$
Pay attention to the parabola of the graph, it does not intersect the x-axis, therefore, the equation has no real solutions.
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
$2{x}^{2}+6x$ =
$0$Coefficient values
$a$ =
$2$$b$ =
$6$An incomplete quadratic equation of the form ax² + bx = 0, where b≠0 has two real solutions:
${x}_{1}$ =
$0$ and
${x}_{2}$ =
$-\frac{b}{\mathrm{}a}$Solution
${x}_{1}$ =
$0$${x}_{2}$ =
$-\frac{6}{\mathrm{}2}$ =
$-3$
Plot of quadratic function$y$ =
$2{x}^{2}+6x$
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}$$c$ =
$-5$An incomplete quadratic equation of the form ax² + c = 0 and has two real solutions if
$\frac{c}{a}<0$ and two complex solutions if
$\frac{c}{a}>0$.
$\frac{c}{a}$ =
$\displaystyle \frac{\mathrm{\left(-5\right)}}{\mathrm{\frac{1}{2}}}$ =
$-10$Solution
$\frac{c}{a}<0$, the equation has two different solutions:
${x}_{1}$ =
$\displaystyle -\sqrt{-\frac{c}{a}}$${x}_{2}$ =
$\displaystyle \sqrt{-\frac{c}{a}}$${x}_{1}$ =
$\displaystyle -\sqrt{-\frac{\mathrm{\left(-5\right)}}{\mathrm{\frac{1}{2}}}}$ =
$-\sqrt{10}$${x}_{2}$ =
$\displaystyle \sqrt{-\frac{\mathrm{\left(-5\right)}}{\mathrm{\frac{1}{2}}}}$ =
$\sqrt{10}$${x}_{1}$ =
$-\sqrt{10}$ =
$-3.16227766016838$${x}_{2}$ =
$\sqrt{10}$ =
$3.16227766016838$
Plot of quadratic function$y$ =
$\frac{{x}^{2}}{2}-5$