Solve for x
x=\frac{\sqrt{17}-1}{4}\approx 0.780776406
x=\frac{-\sqrt{17}-1}{4}\approx -1.280776406
Graph
Share
Copied to clipboard
2x^{2}+x-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{-1±\sqrt{1^{2}-4\times 2\left(-2\right)}}{2\times 2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 1 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-1±\sqrt{1-4\times 2\left(-2\right)}}{2\times 2}
Square 1.
x=\frac{-1±\sqrt{1-8\left(-2\right)}}{2\times 2}
Multiply -4 times 2.
x=\frac{-1±\sqrt{1+16}}{2\times 2}
Multiply -8 times -2.
x=\frac{-1±\sqrt{17}}{2\times 2}
Add 1 to 16.
x=\frac{-1±\sqrt{17}}{4}
Multiply 2 times 2.
x=\frac{\sqrt{17}-1}{4}
Now solve the equation x=\frac{-1±\sqrt{17}}{4} when ± is plus. Add -1 to \sqrt{17}.
x=\frac{-\sqrt{17}-1}{4}
Now solve the equation x=\frac{-1±\sqrt{17}}{4} when ± is minus. Subtract \sqrt{17} from -1.
x=\frac{\sqrt{17}-1}{4} x=\frac{-\sqrt{17}-1}{4}
The equation is now solved.
2x^{2}+x-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.
2x^{2}+x-2-\left(-2\right)=-\left(-2\right)
Add 2 to both sides of the equation.
2x^{2}+x=-\left(-2\right)
Subtracting -2 from itself leaves 0.
2x^{2}+x=2
Subtract -2 from 0.
\frac{2x^{2}+x}{2}=\frac{2}{2}
Divide both sides by 2.
x^{2}+\frac{1}{2}x=\frac{2}{2}
Dividing by 2 undoes the multiplication by 2.
x^{2}+\frac{1}{2}x=1
Divide 2 by 2.
x^{2}+\frac{1}{2}x+\left(\frac{1}{4}\right)^{2}=1+\left(\frac{1}{4}\right)^{2}
Divide \frac{1}{2}, the coefficient of the x term, by 2 to get \frac{1}{4}. Then add the square of \frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{1}{2}x+\frac{1}{16}=1+\frac{1}{16}
Square \frac{1}{4} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{1}{2}x+\frac{1}{16}=\frac{17}{16}
Add 1 to \frac{1}{16}.
\left(x+\frac{1}{4}\right)^{2}=\frac{17}{16}
Factor x^{2}+\frac{1}{2}x+\frac{1}{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{1}{4}\right)^{2}}=\sqrt{\frac{17}{16}}
Take the square root of both sides of the equation.
x+\frac{1}{4}=\frac{\sqrt{17}}{4} x+\frac{1}{4}=-\frac{\sqrt{17}}{4}
Simplify.
x=\frac{\sqrt{17}-1}{4} x=\frac{-\sqrt{17}-1}{4}
Subtract \frac{1}{4} from both sides of the equation.
x ^ 2 +\frac{1}{2}x -1 = 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 2
r + s = -\frac{1}{2} rs = -1
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{1}{4} - u s = -\frac{1}{4} + u
Two numbers r and s sum up to -\frac{1}{2} exactly when the average of the two numbers is \frac{1}{2}*-\frac{1}{2} = -\frac{1}{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{1}{4} - u) (-\frac{1}{4} + u) = -1
To solve for unknown quantity u, substitute these in the product equation rs = -1
\frac{1}{16} - u^2 = -1
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -1-\frac{1}{16} = -\frac{17}{16}
Simplify the expression by subtracting \frac{1}{16} on both sides
u^2 = \frac{17}{16} u = \pm\sqrt{\frac{17}{16}} = \pm \frac{\sqrt{17}}{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{1}{4} - \frac{\sqrt{17}}{4} = -1.281 s = -\frac{1}{4} + \frac{\sqrt{17}}{4} = 0.781
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
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}