Solve for x
x=\frac{\sqrt{46}-8}{3}\approx -0.405890006
x=\frac{-\sqrt{46}-8}{3}\approx -4.927443328
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3x^{2}+16x+6=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{-16±\sqrt{16^{2}-4\times 3\times 6}}{2\times 3}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 16 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-16±\sqrt{256-4\times 3\times 6}}{2\times 3}
Square 16.
x=\frac{-16±\sqrt{256-12\times 6}}{2\times 3}
Multiply -4 times 3.
x=\frac{-16±\sqrt{256-72}}{2\times 3}
Multiply -12 times 6.
x=\frac{-16±\sqrt{184}}{2\times 3}
Add 256 to -72.
x=\frac{-16±2\sqrt{46}}{2\times 3}
Take the square root of 184.
x=\frac{-16±2\sqrt{46}}{6}
Multiply 2 times 3.
x=\frac{2\sqrt{46}-16}{6}
Now solve the equation x=\frac{-16±2\sqrt{46}}{6} when ± is plus. Add -16 to 2\sqrt{46}.
x=\frac{\sqrt{46}-8}{3}
Divide -16+2\sqrt{46} by 6.
x=\frac{-2\sqrt{46}-16}{6}
Now solve the equation x=\frac{-16±2\sqrt{46}}{6} when ± is minus. Subtract 2\sqrt{46} from -16.
x=\frac{-\sqrt{46}-8}{3}
Divide -16-2\sqrt{46} by 6.
x=\frac{\sqrt{46}-8}{3} x=\frac{-\sqrt{46}-8}{3}
The equation is now solved.
3x^{2}+16x+6=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.
3x^{2}+16x+6-6=-6
Subtract 6 from both sides of the equation.
3x^{2}+16x=-6
Subtracting 6 from itself leaves 0.
\frac{3x^{2}+16x}{3}=-\frac{6}{3}
Divide both sides by 3.
x^{2}+\frac{16}{3}x=-\frac{6}{3}
Dividing by 3 undoes the multiplication by 3.
x^{2}+\frac{16}{3}x=-2
Divide -6 by 3.
x^{2}+\frac{16}{3}x+\left(\frac{8}{3}\right)^{2}=-2+\left(\frac{8}{3}\right)^{2}
Divide \frac{16}{3}, the coefficient of the x term, by 2 to get \frac{8}{3}. Then add the square of \frac{8}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{16}{3}x+\frac{64}{9}=-2+\frac{64}{9}
Square \frac{8}{3} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{16}{3}x+\frac{64}{9}=\frac{46}{9}
Add -2 to \frac{64}{9}.
\left(x+\frac{8}{3}\right)^{2}=\frac{46}{9}
Factor x^{2}+\frac{16}{3}x+\frac{64}{9}. 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{8}{3}\right)^{2}}=\sqrt{\frac{46}{9}}
Take the square root of both sides of the equation.
x+\frac{8}{3}=\frac{\sqrt{46}}{3} x+\frac{8}{3}=-\frac{\sqrt{46}}{3}
Simplify.
x=\frac{\sqrt{46}-8}{3} x=\frac{-\sqrt{46}-8}{3}
Subtract \frac{8}{3} from both sides of the equation.
x ^ 2 +\frac{16}{3}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.This is achieved by dividing both sides of the equation by 3
r + s = -\frac{16}{3} 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{8}{3} - u s = -\frac{8}{3} + u
Two numbers r and s sum up to -\frac{16}{3} exactly when the average of the two numbers is \frac{1}{2}*-\frac{16}{3} = -\frac{8}{3}. 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{8}{3} - u) (-\frac{8}{3} + u) = 2
To solve for unknown quantity u, substitute these in the product equation rs = 2
\frac{64}{9} - u^2 = 2
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 2-\frac{64}{9} = -\frac{46}{9}
Simplify the expression by subtracting \frac{64}{9} on both sides
u^2 = \frac{46}{9} u = \pm\sqrt{\frac{46}{9}} = \pm \frac{\sqrt{46}}{3}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-\frac{8}{3} - \frac{\sqrt{46}}{3} = -4.927 s = -\frac{8}{3} + \frac{\sqrt{46}}{3} = -0.406
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
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}