Solve for x
x=\frac{\sqrt{7}-2}{3}\approx 0.215250437
x=\frac{-\sqrt{7}-2}{3}\approx -1.54858377
Graph
Share
Copied to clipboard
3x^{2}+4x+6=7
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.
3x^{2}+4x+6-7=7-7
Subtract 7 from both sides of the equation.
3x^{2}+4x+6-7=0
Subtracting 7 from itself leaves 0.
3x^{2}+4x-1=0
Subtract 7 from 6.
x=\frac{-4±\sqrt{4^{2}-4\times 3\left(-1\right)}}{2\times 3}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 4 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-4±\sqrt{16-4\times 3\left(-1\right)}}{2\times 3}
Square 4.
x=\frac{-4±\sqrt{16-12\left(-1\right)}}{2\times 3}
Multiply -4 times 3.
x=\frac{-4±\sqrt{16+12}}{2\times 3}
Multiply -12 times -1.
x=\frac{-4±\sqrt{28}}{2\times 3}
Add 16 to 12.
x=\frac{-4±2\sqrt{7}}{2\times 3}
Take the square root of 28.
x=\frac{-4±2\sqrt{7}}{6}
Multiply 2 times 3.
x=\frac{2\sqrt{7}-4}{6}
Now solve the equation x=\frac{-4±2\sqrt{7}}{6} when ± is plus. Add -4 to 2\sqrt{7}.
x=\frac{\sqrt{7}-2}{3}
Divide -4+2\sqrt{7} by 6.
x=\frac{-2\sqrt{7}-4}{6}
Now solve the equation x=\frac{-4±2\sqrt{7}}{6} when ± is minus. Subtract 2\sqrt{7} from -4.
x=\frac{-\sqrt{7}-2}{3}
Divide -4-2\sqrt{7} by 6.
x=\frac{\sqrt{7}-2}{3} x=\frac{-\sqrt{7}-2}{3}
The equation is now solved.
3x^{2}+4x+6=7
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}+4x+6-6=7-6
Subtract 6 from both sides of the equation.
3x^{2}+4x=7-6
Subtracting 6 from itself leaves 0.
3x^{2}+4x=1
Subtract 6 from 7.
\frac{3x^{2}+4x}{3}=\frac{1}{3}
Divide both sides by 3.
x^{2}+\frac{4}{3}x=\frac{1}{3}
Dividing by 3 undoes the multiplication by 3.
x^{2}+\frac{4}{3}x+\left(\frac{2}{3}\right)^{2}=\frac{1}{3}+\left(\frac{2}{3}\right)^{2}
Divide \frac{4}{3}, the coefficient of the x term, by 2 to get \frac{2}{3}. Then add the square of \frac{2}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{4}{3}x+\frac{4}{9}=\frac{1}{3}+\frac{4}{9}
Square \frac{2}{3} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{4}{3}x+\frac{4}{9}=\frac{7}{9}
Add \frac{1}{3} to \frac{4}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{2}{3}\right)^{2}=\frac{7}{9}
Factor x^{2}+\frac{4}{3}x+\frac{4}{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{2}{3}\right)^{2}}=\sqrt{\frac{7}{9}}
Take the square root of both sides of the equation.
x+\frac{2}{3}=\frac{\sqrt{7}}{3} x+\frac{2}{3}=-\frac{\sqrt{7}}{3}
Simplify.
x=\frac{\sqrt{7}-2}{3} x=\frac{-\sqrt{7}-2}{3}
Subtract \frac{2}{3} from both sides of the equation.
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}