Solve for x
x = \frac{\sqrt{57} + 5}{4} \approx 3.137458609
x=\frac{5-\sqrt{57}}{4}\approx -0.637458609
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-2x^{2}+5x+4=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\left(-2\right)\times 4}}{2\left(-2\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 5 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-5±\sqrt{25-4\left(-2\right)\times 4}}{2\left(-2\right)}
Square 5.
x=\frac{-5±\sqrt{25+8\times 4}}{2\left(-2\right)}
Multiply -4 times -2.
x=\frac{-5±\sqrt{25+32}}{2\left(-2\right)}
Multiply 8 times 4.
x=\frac{-5±\sqrt{57}}{2\left(-2\right)}
Add 25 to 32.
x=\frac{-5±\sqrt{57}}{-4}
Multiply 2 times -2.
x=\frac{\sqrt{57}-5}{-4}
Now solve the equation x=\frac{-5±\sqrt{57}}{-4} when ± is plus. Add -5 to \sqrt{57}.
x=\frac{5-\sqrt{57}}{4}
Divide -5+\sqrt{57} by -4.
x=\frac{-\sqrt{57}-5}{-4}
Now solve the equation x=\frac{-5±\sqrt{57}}{-4} when ± is minus. Subtract \sqrt{57} from -5.
x=\frac{\sqrt{57}+5}{4}
Divide -5-\sqrt{57} by -4.
x=\frac{5-\sqrt{57}}{4} x=\frac{\sqrt{57}+5}{4}
The equation is now solved.
-2x^{2}+5x+4=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.
-2x^{2}+5x+4-4=-4
Subtract 4 from both sides of the equation.
-2x^{2}+5x=-4
Subtracting 4 from itself leaves 0.
\frac{-2x^{2}+5x}{-2}=-\frac{4}{-2}
Divide both sides by -2.
x^{2}+\frac{5}{-2}x=-\frac{4}{-2}
Dividing by -2 undoes the multiplication by -2.
x^{2}-\frac{5}{2}x=-\frac{4}{-2}
Divide 5 by -2.
x^{2}-\frac{5}{2}x=2
Divide -4 by -2.
x^{2}-\frac{5}{2}x+\left(-\frac{5}{4}\right)^{2}=2+\left(-\frac{5}{4}\right)^{2}
Divide -\frac{5}{2}, the coefficient of the x term, by 2 to get -\frac{5}{4}. Then add the square of -\frac{5}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{5}{2}x+\frac{25}{16}=2+\frac{25}{16}
Square -\frac{5}{4} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{5}{2}x+\frac{25}{16}=\frac{57}{16}
Add 2 to \frac{25}{16}.
\left(x-\frac{5}{4}\right)^{2}=\frac{57}{16}
Factor x^{2}-\frac{5}{2}x+\frac{25}{16}. 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}{4}\right)^{2}}=\sqrt{\frac{57}{16}}
Take the square root of both sides of the equation.
x-\frac{5}{4}=\frac{\sqrt{57}}{4} x-\frac{5}{4}=-\frac{\sqrt{57}}{4}
Simplify.
x=\frac{\sqrt{57}+5}{4} x=\frac{5-\sqrt{57}}{4}
Add \frac{5}{4} to both sides of the equation.
x ^ 2 -\frac{5}{2}x -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.
r + s = \frac{5}{2} rs = -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}{4} - u s = \frac{5}{4} + u
Two numbers r and s sum up to \frac{5}{2} exactly when the average of the two numbers is \frac{1}{2}*\frac{5}{2} = \frac{5}{4}. 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}{4} - u) (\frac{5}{4} + u) = -2
To solve for unknown quantity u, substitute these in the product equation rs = -2
\frac{25}{16} - u^2 = -2
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -2-\frac{25}{16} = -\frac{57}{16}
Simplify the expression by subtracting \frac{25}{16} on both sides
u^2 = \frac{57}{16} u = \pm\sqrt{\frac{57}{16}} = \pm \frac{\sqrt{57}}{4}
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}{4} - \frac{\sqrt{57}}{4} = -0.637 s = \frac{5}{4} + \frac{\sqrt{57}}{4} = 3.137
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
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Simultaneous equation
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Differentiation
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Integration
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Limits
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