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a+b=11 ab=21\left(-2\right)=-42
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 21x^{2}+ax+bx-2. To find a and b, set up a system to be solved.
-1,42 -2,21 -3,14 -6,7
Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -42.
-1+42=41 -2+21=19 -3+14=11 -6+7=1
Calculate the sum for each pair.
a=-3 b=14
The solution is the pair that gives sum 11.
\left(21x^{2}-3x\right)+\left(14x-2\right)
Rewrite 21x^{2}+11x-2 as \left(21x^{2}-3x\right)+\left(14x-2\right).
3x\left(7x-1\right)+2\left(7x-1\right)
Factor out 3x in the first and 2 in the second group.
\left(7x-1\right)\left(3x+2\right)
Factor out common term 7x-1 by using distributive property.
x=\frac{1}{7} x=-\frac{2}{3}
To find equation solutions, solve 7x-1=0 and 3x+2=0.
21x^{2}+11x-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{-11±\sqrt{11^{2}-4\times 21\left(-2\right)}}{2\times 21}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 21 for a, 11 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-11±\sqrt{121-4\times 21\left(-2\right)}}{2\times 21}
Square 11.
x=\frac{-11±\sqrt{121-84\left(-2\right)}}{2\times 21}
Multiply -4 times 21.
x=\frac{-11±\sqrt{121+168}}{2\times 21}
Multiply -84 times -2.
x=\frac{-11±\sqrt{289}}{2\times 21}
Add 121 to 168.
x=\frac{-11±17}{2\times 21}
Take the square root of 289.
x=\frac{-11±17}{42}
Multiply 2 times 21.
x=\frac{6}{42}
Now solve the equation x=\frac{-11±17}{42} when ± is plus. Add -11 to 17.
x=\frac{1}{7}
Reduce the fraction \frac{6}{42} to lowest terms by extracting and canceling out 6.
x=-\frac{28}{42}
Now solve the equation x=\frac{-11±17}{42} when ± is minus. Subtract 17 from -11.
x=-\frac{2}{3}
Reduce the fraction \frac{-28}{42} to lowest terms by extracting and canceling out 14.
x=\frac{1}{7} x=-\frac{2}{3}
The equation is now solved.
21x^{2}+11x-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.
21x^{2}+11x-2-\left(-2\right)=-\left(-2\right)
Add 2 to both sides of the equation.
21x^{2}+11x=-\left(-2\right)
Subtracting -2 from itself leaves 0.
21x^{2}+11x=2
Subtract -2 from 0.
\frac{21x^{2}+11x}{21}=\frac{2}{21}
Divide both sides by 21.
x^{2}+\frac{11}{21}x=\frac{2}{21}
Dividing by 21 undoes the multiplication by 21.
x^{2}+\frac{11}{21}x+\left(\frac{11}{42}\right)^{2}=\frac{2}{21}+\left(\frac{11}{42}\right)^{2}
Divide \frac{11}{21}, the coefficient of the x term, by 2 to get \frac{11}{42}. Then add the square of \frac{11}{42} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{11}{21}x+\frac{121}{1764}=\frac{2}{21}+\frac{121}{1764}
Square \frac{11}{42} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{11}{21}x+\frac{121}{1764}=\frac{289}{1764}
Add \frac{2}{21} to \frac{121}{1764} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{11}{42}\right)^{2}=\frac{289}{1764}
Factor x^{2}+\frac{11}{21}x+\frac{121}{1764}. 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{11}{42}\right)^{2}}=\sqrt{\frac{289}{1764}}
Take the square root of both sides of the equation.
x+\frac{11}{42}=\frac{17}{42} x+\frac{11}{42}=-\frac{17}{42}
Simplify.
x=\frac{1}{7} x=-\frac{2}{3}
Subtract \frac{11}{42} from both sides of the equation.
x ^ 2 +\frac{11}{21}x -\frac{2}{21} = 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 21
r + s = -\frac{11}{21} rs = -\frac{2}{21}
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{11}{42} - u s = -\frac{11}{42} + u
Two numbers r and s sum up to -\frac{11}{21} exactly when the average of the two numbers is \frac{1}{2}*-\frac{11}{21} = -\frac{11}{42}. 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{11}{42} - u) (-\frac{11}{42} + u) = -\frac{2}{21}
To solve for unknown quantity u, substitute these in the product equation rs = -\frac{2}{21}
\frac{121}{1764} - u^2 = -\frac{2}{21}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -\frac{2}{21}-\frac{121}{1764} = -\frac{289}{1764}
Simplify the expression by subtracting \frac{121}{1764} on both sides
u^2 = \frac{289}{1764} u = \pm\sqrt{\frac{289}{1764}} = \pm \frac{17}{42}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-\frac{11}{42} - \frac{17}{42} = -0.667 s = -\frac{11}{42} + \frac{17}{42} = 0.143
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.