Solve for x (complex solution)
x=\frac{-5+\sqrt{7}i}{8}\approx -0.625+0.330718914i
x=\frac{-\sqrt{7}i-5}{8}\approx -0.625-0.330718914i
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4x^{2}+5x+2=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{-5±\sqrt{5^{2}-4\times 4\times 2}}{2\times 4}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 5 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-5±\sqrt{25-4\times 4\times 2}}{2\times 4}
Square 5.
x=\frac{-5±\sqrt{25-16\times 2}}{2\times 4}
Multiply -4 times 4.
x=\frac{-5±\sqrt{25-32}}{2\times 4}
Multiply -16 times 2.
x=\frac{-5±\sqrt{-7}}{2\times 4}
Add 25 to -32.
x=\frac{-5±\sqrt{7}i}{2\times 4}
Take the square root of -7.
x=\frac{-5±\sqrt{7}i}{8}
Multiply 2 times 4.
x=\frac{-5+\sqrt{7}i}{8}
Now solve the equation x=\frac{-5±\sqrt{7}i}{8} when ± is plus. Add -5 to i\sqrt{7}.
x=\frac{-\sqrt{7}i-5}{8}
Now solve the equation x=\frac{-5±\sqrt{7}i}{8} when ± is minus. Subtract i\sqrt{7} from -5.
x=\frac{-5+\sqrt{7}i}{8} x=\frac{-\sqrt{7}i-5}{8}
The equation is now solved.
4x^{2}+5x+2=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.
4x^{2}+5x+2-2=-2
Subtract 2 from both sides of the equation.
4x^{2}+5x=-2
Subtracting 2 from itself leaves 0.
\frac{4x^{2}+5x}{4}=-\frac{2}{4}
Divide both sides by 4.
x^{2}+\frac{5}{4}x=-\frac{2}{4}
Dividing by 4 undoes the multiplication by 4.
x^{2}+\frac{5}{4}x=-\frac{1}{2}
Reduce the fraction \frac{-2}{4} to lowest terms by extracting and canceling out 2.
x^{2}+\frac{5}{4}x+\left(\frac{5}{8}\right)^{2}=-\frac{1}{2}+\left(\frac{5}{8}\right)^{2}
Divide \frac{5}{4}, the coefficient of the x term, by 2 to get \frac{5}{8}. Then add the square of \frac{5}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{5}{4}x+\frac{25}{64}=-\frac{1}{2}+\frac{25}{64}
Square \frac{5}{8} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{5}{4}x+\frac{25}{64}=-\frac{7}{64}
Add -\frac{1}{2} to \frac{25}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{5}{8}\right)^{2}=-\frac{7}{64}
Factor x^{2}+\frac{5}{4}x+\frac{25}{64}. 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{5}{8}\right)^{2}}=\sqrt{-\frac{7}{64}}
Take the square root of both sides of the equation.
x+\frac{5}{8}=\frac{\sqrt{7}i}{8} x+\frac{5}{8}=-\frac{\sqrt{7}i}{8}
Simplify.
x=\frac{-5+\sqrt{7}i}{8} x=\frac{-\sqrt{7}i-5}{8}
Subtract \frac{5}{8} from both sides of the equation.
x ^ 2 +\frac{5}{4}x +\frac{1}{2} = 0
Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.This is achieved by dividing both sides of the equation by 4
r + s = -\frac{5}{4} rs = \frac{1}{2}
Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C
r = -\frac{5}{8} - u s = -\frac{5}{8} + u
Two numbers r and s sum up to -\frac{5}{4} exactly when the average of the two numbers is \frac{1}{2}*-\frac{5}{4} = -\frac{5}{8}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>
(-\frac{5}{8} - u) (-\frac{5}{8} + u) = \frac{1}{2}
To solve for unknown quantity u, substitute these in the product equation rs = \frac{1}{2}
\frac{25}{64} - u^2 = \frac{1}{2}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = \frac{1}{2}-\frac{25}{64} = \frac{7}{64}
Simplify the expression by subtracting \frac{25}{64} on both sides
u^2 = -\frac{7}{64} u = \pm\sqrt{-\frac{7}{64}} = \pm \frac{\sqrt{7}}{8}i
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-\frac{5}{8} - \frac{\sqrt{7}}{8}i = -0.625 - 0.331i s = -\frac{5}{8} + \frac{\sqrt{7}}{8}i = -0.625 + 0.331i
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
Examples
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{ x } ^ { 2 } - 4 x - 5 = 0
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4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
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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
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