Solve for x
x = \frac{\sqrt{2006} + 12}{49} \approx 1.158946762
x=\frac{12-\sqrt{2006}}{49}\approx -0.669150844
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4.9x^{2}-2.4x-3.8=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{-\left(-2.4\right)±\sqrt{\left(-2.4\right)^{2}-4\times 4.9\left(-3.8\right)}}{2\times 4.9}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 4.9 for a, -2.4 for b, and -3.8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-2.4\right)±\sqrt{5.76-4\times 4.9\left(-3.8\right)}}{2\times 4.9}
Square -2.4 by squaring both the numerator and the denominator of the fraction.
x=\frac{-\left(-2.4\right)±\sqrt{5.76-19.6\left(-3.8\right)}}{2\times 4.9}
Multiply -4 times 4.9.
x=\frac{-\left(-2.4\right)±\sqrt{\frac{144+1862}{25}}}{2\times 4.9}
Multiply -19.6 times -3.8 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.
x=\frac{-\left(-2.4\right)±\sqrt{80.24}}{2\times 4.9}
Add 5.76 to 74.48 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
x=\frac{-\left(-2.4\right)±\frac{\sqrt{2006}}{5}}{2\times 4.9}
Take the square root of 80.24.
x=\frac{2.4±\frac{\sqrt{2006}}{5}}{2\times 4.9}
The opposite of -2.4 is 2.4.
x=\frac{2.4±\frac{\sqrt{2006}}{5}}{9.8}
Multiply 2 times 4.9.
x=\frac{\sqrt{2006}+12}{5\times 9.8}
Now solve the equation x=\frac{2.4±\frac{\sqrt{2006}}{5}}{9.8} when ± is plus. Add 2.4 to \frac{\sqrt{2006}}{5}.
x=\frac{\sqrt{2006}+12}{49}
Divide \frac{12+\sqrt{2006}}{5} by 9.8 by multiplying \frac{12+\sqrt{2006}}{5} by the reciprocal of 9.8.
x=\frac{12-\sqrt{2006}}{5\times 9.8}
Now solve the equation x=\frac{2.4±\frac{\sqrt{2006}}{5}}{9.8} when ± is minus. Subtract \frac{\sqrt{2006}}{5} from 2.4.
x=\frac{12-\sqrt{2006}}{49}
Divide \frac{12-\sqrt{2006}}{5} by 9.8 by multiplying \frac{12-\sqrt{2006}}{5} by the reciprocal of 9.8.
x=\frac{\sqrt{2006}+12}{49} x=\frac{12-\sqrt{2006}}{49}
The equation is now solved.
4.9x^{2}-2.4x-3.8=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.
4.9x^{2}-2.4x-3.8-\left(-3.8\right)=-\left(-3.8\right)
Add 3.8 to both sides of the equation.
4.9x^{2}-2.4x=-\left(-3.8\right)
Subtracting -3.8 from itself leaves 0.
4.9x^{2}-2.4x=3.8
Subtract -3.8 from 0.
\frac{4.9x^{2}-2.4x}{4.9}=\frac{3.8}{4.9}
Divide both sides of the equation by 4.9, which is the same as multiplying both sides by the reciprocal of the fraction.
x^{2}+\left(-\frac{2.4}{4.9}\right)x=\frac{3.8}{4.9}
Dividing by 4.9 undoes the multiplication by 4.9.
x^{2}-\frac{24}{49}x=\frac{3.8}{4.9}
Divide -2.4 by 4.9 by multiplying -2.4 by the reciprocal of 4.9.
x^{2}-\frac{24}{49}x=\frac{38}{49}
Divide 3.8 by 4.9 by multiplying 3.8 by the reciprocal of 4.9.
x^{2}-\frac{24}{49}x+\left(-\frac{12}{49}\right)^{2}=\frac{38}{49}+\left(-\frac{12}{49}\right)^{2}
Divide -\frac{24}{49}, the coefficient of the x term, by 2 to get -\frac{12}{49}. Then add the square of -\frac{12}{49} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{24}{49}x+\frac{144}{2401}=\frac{38}{49}+\frac{144}{2401}
Square -\frac{12}{49} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{24}{49}x+\frac{144}{2401}=\frac{2006}{2401}
Add \frac{38}{49} to \frac{144}{2401} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{12}{49}\right)^{2}=\frac{2006}{2401}
Factor x^{2}-\frac{24}{49}x+\frac{144}{2401}. 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{12}{49}\right)^{2}}=\sqrt{\frac{2006}{2401}}
Take the square root of both sides of the equation.
x-\frac{12}{49}=\frac{\sqrt{2006}}{49} x-\frac{12}{49}=-\frac{\sqrt{2006}}{49}
Simplify.
x=\frac{\sqrt{2006}+12}{49} x=\frac{12-\sqrt{2006}}{49}
Add \frac{12}{49} to both sides of the equation.
Examples
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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}