Solve for a (complex solution)

\left\{\begin{matrix}a=-\frac{bx+c}{x^{2}}\text{, }&x\neq 0\\a\in \mathrm{C}\text{, }&c=0\text{ and }x=0\end{matrix}\right.

${a=−x_{2}bx+c ,a∈C, x =0c=0andx=0 $

Solve for b (complex solution)

\left\{\begin{matrix}b=-ax-\frac{c}{x}\text{, }&x\neq 0\\b\in \mathrm{C}\text{, }&c=0\text{ and }x=0\end{matrix}\right.

${b=−ax−xc ,b∈C, x =0c=0andx=0 $

Solve for a

\left\{\begin{matrix}a=-\frac{bx+c}{x^{2}}\text{, }&x\neq 0\\a\in \mathrm{R}\text{, }&c=0\text{ and }x=0\end{matrix}\right.

${a=−x_{2}bx+c ,a∈R, x =0c=0andx=0 $

Solve for b

\left\{\begin{matrix}b=-ax-\frac{c}{x}\text{, }&x\neq 0\\b\in \mathrm{R}\text{, }&c=0\text{ and }x=0\end{matrix}\right.

${b=−ax−xc ,b∈R, x =0c=0andx=0 $

Solve for x

\left\{\begin{matrix}x=\frac{\sqrt{b^{2}-4ac}-b}{2a}\text{; }x=-\frac{\sqrt{b^{2}-4ac}+b}{2a}\text{, }&\left(b\neq 0\text{ and }a=\frac{b^{2}}{4c}\text{ and }c\neq 0\right)\text{ or }\left(a\neq 0\text{ and }a\geq \frac{b^{2}}{4c}\text{ and }c\leq 0\right)\text{ or }\left(c=0\text{ and }a\neq 0\right)\text{ or }\left(a\neq 0\text{ and }c\geq 0\text{ and }a\leq \frac{b^{2}}{4c}\right)\\x=-\frac{c}{b}\text{, }&a=0\text{ and }b\neq 0\\x\in \mathrm{R}\text{, }&a=0\text{ and }b=0\text{ and }c=0\end{matrix}\right.

$⎩⎪⎪⎨⎪⎪⎧ x=2ab_{2}−4ac −b ;x=−2ab_{2}−4ac +b ,x=−bc ,x∈R, (b =0anda=4cb_{2} andc =0)or(a =0anda≥4cb_{2} andc≤0)or(c=0anda =0)or(a =0andc≥0anda≤4cb_{2} )a=0andb =0a=0andb=0andc=0 $

Steps Using the Quadratic Formula

Steps for Completing the Square

Solve for x (complex solution)

\left\{\begin{matrix}x=\frac{\sqrt{b^{2}-4ac}-b}{2a}\text{; }x=-\frac{\sqrt{b^{2}-4ac}+b}{2a}\text{, }&a\neq 0\\x=-\frac{c}{b}\text{, }&a=0\text{ and }b\neq 0\\x\in \mathrm{C}\text{, }&a=0\text{ and }b=0\text{ and }c=0\end{matrix}\right.

$⎩⎪⎨⎪⎧ x=2ab_{2}−4ac −b ;x=−2ab_{2}−4ac +b ,x=−bc ,x∈C, a =0a=0andb =0a=0andb=0andc=0 $

Solve for c

c=-x\left(ax+b\right)

$c=−x(ax+b)$

Graph

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ax^{2}+c=-bx

Subtract bx from both sides. Anything subtracted from zero gives its negation.

ax^{2}=-bx-c

Subtract c from both sides.

x^{2}a=-bx-c

The equation is in standard form.

\frac{x^{2}a}{x^{2}}=\frac{-bx-c}{x^{2}}

Divide both sides by x^{2}.

a=\frac{-bx-c}{x^{2}}

Dividing by x^{2} undoes the multiplication by x^{2}.

a=-\frac{bx+c}{x^{2}}

Divide -bx-c by x^{2}.

bx+c=-ax^{2}

Subtract ax^{2} from both sides. Anything subtracted from zero gives its negation.

bx=-ax^{2}-c

Subtract c from both sides.

xb=-ax^{2}-c

The equation is in standard form.

\frac{xb}{x}=\frac{-ax^{2}-c}{x}

Divide both sides by x.

b=\frac{-ax^{2}-c}{x}

Dividing by x undoes the multiplication by x.

b=-ax-\frac{c}{x}

Divide -ax^{2}-c by x.

ax^{2}+c=-bx

Subtract bx from both sides. Anything subtracted from zero gives its negation.

ax^{2}=-bx-c

Subtract c from both sides.

x^{2}a=-bx-c

The equation is in standard form.

\frac{x^{2}a}{x^{2}}=\frac{-bx-c}{x^{2}}

Divide both sides by x^{2}.

a=\frac{-bx-c}{x^{2}}

Dividing by x^{2} undoes the multiplication by x^{2}.

a=-\frac{bx+c}{x^{2}}

Divide -bx-c by x^{2}.

bx+c=-ax^{2}

Subtract ax^{2} from both sides. Anything subtracted from zero gives its negation.

bx=-ax^{2}-c

Subtract c from both sides.

xb=-ax^{2}-c

The equation is in standard form.

\frac{xb}{x}=\frac{-ax^{2}-c}{x}

Divide both sides by x.

b=\frac{-ax^{2}-c}{x}

Dividing by x undoes the multiplication by x.

b=-ax-\frac{c}{x}

Divide -ax^{2}-c by x.

ax^{2}+bx+c=0

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

x=\frac{-b±\sqrt{b^{2}-4ac}}{2a}

This equation is in standard form: ax^{2}+bx+c=0. Substitute a for a, b for b, and c for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

x=\frac{\sqrt{b^{2}-4ac}-b}{2a}

Now solve the equation x=\frac{-b±\sqrt{b^{2}-4ac}}{2a} when ± is plus. Add -b to \sqrt{b^{2}-4ac}.

x=\frac{-\sqrt{b^{2}-4ac}-b}{2a}

Now solve the equation x=\frac{-b±\sqrt{b^{2}-4ac}}{2a} when ± is minus. Subtract \sqrt{b^{2}-4ac} from -b.

x=-\frac{\sqrt{b^{2}-4ac}+b}{2a}

Divide -b-\sqrt{b^{2}-4ac} by 2a.

x=\frac{\sqrt{b^{2}-4ac}-b}{2a} x=-\frac{\sqrt{b^{2}-4ac}+b}{2a}

The equation is now solved.

ax^{2}+bx+c=0

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

ax^{2}+bx+c-c=-c

Subtract c from both sides of the equation.

ax^{2}+bx=-c

Subtracting c from itself leaves 0.

\frac{ax^{2}+bx}{a}=-\frac{c}{a}

Divide both sides by a.

x^{2}+\frac{b}{a}x=-\frac{c}{a}

Dividing by a undoes the multiplication by a.

x^{2}+\frac{b}{a}x+\left(\frac{b}{2a}\right)^{2}=-\frac{c}{a}+\left(\frac{b}{2a}\right)^{2}

Divide \frac{b}{a}, the coefficient of the x term, by 2 to get \frac{b}{2a}. Then add the square of \frac{b}{2a} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

x^{2}+\frac{b}{a}x+\frac{b^{2}}{4a^{2}}=-\frac{c}{a}+\frac{b^{2}}{4a^{2}}

Square \frac{b}{2a}.

x^{2}+\frac{b}{a}x+\frac{b^{2}}{4a^{2}}=\frac{b^{2}-4ac}{4a^{2}}

Add -\frac{c}{a} to \frac{b^{2}}{4a^{2}}.

\left(x+\frac{b}{2a}\right)^{2}=\frac{b^{2}-4ac}{4a^{2}}

Factor x^{2}+\frac{b}{a}x+\frac{b^{2}}{4a^{2}}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

\sqrt{\left(x+\frac{b}{2a}\right)^{2}}=\sqrt{\frac{b^{2}-4ac}{4a^{2}}}

Take the square root of both sides of the equation.

x+\frac{b}{2a}=\frac{\sqrt{b^{2}-4ac}}{2|a|} x+\frac{b}{2a}=-\frac{\sqrt{b^{2}-4ac}}{2|a|}

Simplify.

x=\frac{\sqrt{b^{2}-4ac}}{2|a|}-\frac{b}{2a} x=-\frac{\sqrt{b^{2}-4ac}}{2|a|}-\frac{b}{2a}

Subtract \frac{b}{2a} from both sides of the equation.

Examples

Quadratic equation

{ x } ^ { 2 } - 4 x - 5 = 0

$x_{2}−4x−5=0$

Trigonometry

4 \sin \theta \cos \theta = 2 \sin \theta

$4sinθcosθ=2sinθ$

Linear equation

y = 3x + 4

$y=3x+4$

Arithmetic

699 * 533

$699∗533$

Matrix

\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { - 1 } & { 1 } & { 5 } \end{array} \right]

$[25 34 ][2−1 01 35 ]$

Simultaneous equation

\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.

${8x+2y=467x+3y=47 $

Differentiation

\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }

$dxd (x−5)(3x_{2}−2) $

Integration

\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x

$∫_{0}xe_{−x_{2}}dx$

Limits

\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}

$x→−3lim x_{2}+2x−3x_{2}−9 $