Evaluate
-6+4i
Real Part
-6
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\frac{-5-5\times \left(-5i\right)-i-\left(-5i^{2}\right)}{3-2i}
Multiply complex numbers -5-i and 1-5i like you multiply binomials.
\frac{-5-5\times \left(-5i\right)-i-\left(-5\left(-1\right)\right)}{3-2i}
By definition, i^{2} is -1.
\frac{-5+25i-i-5}{3-2i}
Do the multiplications in -5-5\times \left(-5i\right)-i-\left(-5\left(-1\right)\right).
\frac{-5-5+\left(25-1\right)i}{3-2i}
Combine the real and imaginary parts in -5+25i-i-5.
\frac{-10+24i}{3-2i}
Do the additions in -5-5+\left(25-1\right)i.
\frac{\left(-10+24i\right)\left(3+2i\right)}{\left(3-2i\right)\left(3+2i\right)}
Multiply both numerator and denominator by the complex conjugate of the denominator, 3+2i.
\frac{\left(-10+24i\right)\left(3+2i\right)}{3^{2}-2^{2}i^{2}}
Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.
\frac{\left(-10+24i\right)\left(3+2i\right)}{13}
By definition, i^{2} is -1. Calculate the denominator.
\frac{-10\times 3-10\times \left(2i\right)+24i\times 3+24\times 2i^{2}}{13}
Multiply complex numbers -10+24i and 3+2i like you multiply binomials.
\frac{-10\times 3-10\times \left(2i\right)+24i\times 3+24\times 2\left(-1\right)}{13}
By definition, i^{2} is -1.
\frac{-30-20i+72i-48}{13}
Do the multiplications in -10\times 3-10\times \left(2i\right)+24i\times 3+24\times 2\left(-1\right).
\frac{-30-48+\left(-20+72\right)i}{13}
Combine the real and imaginary parts in -30-20i+72i-48.
\frac{-78+52i}{13}
Do the additions in -30-48+\left(-20+72\right)i.
-6+4i
Divide -78+52i by 13 to get -6+4i.
Re(\frac{-5-5\times \left(-5i\right)-i-\left(-5i^{2}\right)}{3-2i})
Multiply complex numbers -5-i and 1-5i like you multiply binomials.
Re(\frac{-5-5\times \left(-5i\right)-i-\left(-5\left(-1\right)\right)}{3-2i})
By definition, i^{2} is -1.
Re(\frac{-5+25i-i-5}{3-2i})
Do the multiplications in -5-5\times \left(-5i\right)-i-\left(-5\left(-1\right)\right).
Re(\frac{-5-5+\left(25-1\right)i}{3-2i})
Combine the real and imaginary parts in -5+25i-i-5.
Re(\frac{-10+24i}{3-2i})
Do the additions in -5-5+\left(25-1\right)i.
Re(\frac{\left(-10+24i\right)\left(3+2i\right)}{\left(3-2i\right)\left(3+2i\right)})
Multiply both numerator and denominator of \frac{-10+24i}{3-2i} by the complex conjugate of the denominator, 3+2i.
Re(\frac{\left(-10+24i\right)\left(3+2i\right)}{3^{2}-2^{2}i^{2}})
Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.
Re(\frac{\left(-10+24i\right)\left(3+2i\right)}{13})
By definition, i^{2} is -1. Calculate the denominator.
Re(\frac{-10\times 3-10\times \left(2i\right)+24i\times 3+24\times 2i^{2}}{13})
Multiply complex numbers -10+24i and 3+2i like you multiply binomials.
Re(\frac{-10\times 3-10\times \left(2i\right)+24i\times 3+24\times 2\left(-1\right)}{13})
By definition, i^{2} is -1.
Re(\frac{-30-20i+72i-48}{13})
Do the multiplications in -10\times 3-10\times \left(2i\right)+24i\times 3+24\times 2\left(-1\right).
Re(\frac{-30-48+\left(-20+72\right)i}{13})
Combine the real and imaginary parts in -30-20i+72i-48.
Re(\frac{-78+52i}{13})
Do the additions in -30-48+\left(-20+72\right)i.
Re(-6+4i)
Divide -78+52i by 13 to get -6+4i.
-6
The real part of -6+4i is -6.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
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]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}