Solve for x
x=-\frac{2}{7}\approx -0.285714286
x=\frac{4}{7}\approx 0.571428571
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49x^{2}-14x-8=0
Subtract 8 from both sides.
a+b=-14 ab=49\left(-8\right)=-392
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 49x^{2}+ax+bx-8. To find a and b, set up a system to be solved.
1,-392 2,-196 4,-98 7,-56 8,-49 14,-28
Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -392.
1-392=-391 2-196=-194 4-98=-94 7-56=-49 8-49=-41 14-28=-14
Calculate the sum for each pair.
a=-28 b=14
The solution is the pair that gives sum -14.
\left(49x^{2}-28x\right)+\left(14x-8\right)
Rewrite 49x^{2}-14x-8 as \left(49x^{2}-28x\right)+\left(14x-8\right).
7x\left(7x-4\right)+2\left(7x-4\right)
Factor out 7x in the first and 2 in the second group.
\left(7x-4\right)\left(7x+2\right)
Factor out common term 7x-4 by using distributive property.
x=\frac{4}{7} x=-\frac{2}{7}
To find equation solutions, solve 7x-4=0 and 7x+2=0.
49x^{2}-14x=8
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.
49x^{2}-14x-8=8-8
Subtract 8 from both sides of the equation.
49x^{2}-14x-8=0
Subtracting 8 from itself leaves 0.
x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 49\left(-8\right)}}{2\times 49}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 49 for a, -14 for b, and -8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-14\right)±\sqrt{196-4\times 49\left(-8\right)}}{2\times 49}
Square -14.
x=\frac{-\left(-14\right)±\sqrt{196-196\left(-8\right)}}{2\times 49}
Multiply -4 times 49.
x=\frac{-\left(-14\right)±\sqrt{196+1568}}{2\times 49}
Multiply -196 times -8.
x=\frac{-\left(-14\right)±\sqrt{1764}}{2\times 49}
Add 196 to 1568.
x=\frac{-\left(-14\right)±42}{2\times 49}
Take the square root of 1764.
x=\frac{14±42}{2\times 49}
The opposite of -14 is 14.
x=\frac{14±42}{98}
Multiply 2 times 49.
x=\frac{56}{98}
Now solve the equation x=\frac{14±42}{98} when ± is plus. Add 14 to 42.
x=\frac{4}{7}
Reduce the fraction \frac{56}{98} to lowest terms by extracting and canceling out 14.
x=-\frac{28}{98}
Now solve the equation x=\frac{14±42}{98} when ± is minus. Subtract 42 from 14.
x=-\frac{2}{7}
Reduce the fraction \frac{-28}{98} to lowest terms by extracting and canceling out 14.
x=\frac{4}{7} x=-\frac{2}{7}
The equation is now solved.
49x^{2}-14x=8
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.
\frac{49x^{2}-14x}{49}=\frac{8}{49}
Divide both sides by 49.
x^{2}+\left(-\frac{14}{49}\right)x=\frac{8}{49}
Dividing by 49 undoes the multiplication by 49.
x^{2}-\frac{2}{7}x=\frac{8}{49}
Reduce the fraction \frac{-14}{49} to lowest terms by extracting and canceling out 7.
x^{2}-\frac{2}{7}x+\left(-\frac{1}{7}\right)^{2}=\frac{8}{49}+\left(-\frac{1}{7}\right)^{2}
Divide -\frac{2}{7}, the coefficient of the x term, by 2 to get -\frac{1}{7}. Then add the square of -\frac{1}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{2}{7}x+\frac{1}{49}=\frac{8+1}{49}
Square -\frac{1}{7} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{2}{7}x+\frac{1}{49}=\frac{9}{49}
Add \frac{8}{49} to \frac{1}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{1}{7}\right)^{2}=\frac{9}{49}
Factor x^{2}-\frac{2}{7}x+\frac{1}{49}. 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}{7}\right)^{2}}=\sqrt{\frac{9}{49}}
Take the square root of both sides of the equation.
x-\frac{1}{7}=\frac{3}{7} x-\frac{1}{7}=-\frac{3}{7}
Simplify.
x=\frac{4}{7} x=-\frac{2}{7}
Add \frac{1}{7} to both sides of the equation.
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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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