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a+b=29 ab=12\times 14=168
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 12x^{2}+ax+bx+14. To find a and b, set up a system to be solved.
1,168 2,84 3,56 4,42 6,28 7,24 8,21 12,14
Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 168.
1+168=169 2+84=86 3+56=59 4+42=46 6+28=34 7+24=31 8+21=29 12+14=26
Calculate the sum for each pair.
a=8 b=21
The solution is the pair that gives sum 29.
\left(12x^{2}+8x\right)+\left(21x+14\right)
Rewrite 12x^{2}+29x+14 as \left(12x^{2}+8x\right)+\left(21x+14\right).
4x\left(3x+2\right)+7\left(3x+2\right)
Factor out 4x in the first and 7 in the second group.
\left(3x+2\right)\left(4x+7\right)
Factor out common term 3x+2 by using distributive property.
x=-\frac{2}{3} x=-\frac{7}{4}
To find equation solutions, solve 3x+2=0 and 4x+7=0.
12x^{2}+29x+14=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{-29±\sqrt{29^{2}-4\times 12\times 14}}{2\times 12}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 12 for a, 29 for b, and 14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-29±\sqrt{841-4\times 12\times 14}}{2\times 12}
Square 29.
x=\frac{-29±\sqrt{841-48\times 14}}{2\times 12}
Multiply -4 times 12.
x=\frac{-29±\sqrt{841-672}}{2\times 12}
Multiply -48 times 14.
x=\frac{-29±\sqrt{169}}{2\times 12}
Add 841 to -672.
x=\frac{-29±13}{2\times 12}
Take the square root of 169.
x=\frac{-29±13}{24}
Multiply 2 times 12.
x=-\frac{16}{24}
Now solve the equation x=\frac{-29±13}{24} when ± is plus. Add -29 to 13.
x=-\frac{2}{3}
Reduce the fraction \frac{-16}{24} to lowest terms by extracting and canceling out 8.
x=-\frac{42}{24}
Now solve the equation x=\frac{-29±13}{24} when ± is minus. Subtract 13 from -29.
x=-\frac{7}{4}
Reduce the fraction \frac{-42}{24} to lowest terms by extracting and canceling out 6.
x=-\frac{2}{3} x=-\frac{7}{4}
The equation is now solved.
12x^{2}+29x+14=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.
12x^{2}+29x+14-14=-14
Subtract 14 from both sides of the equation.
12x^{2}+29x=-14
Subtracting 14 from itself leaves 0.
\frac{12x^{2}+29x}{12}=-\frac{14}{12}
Divide both sides by 12.
x^{2}+\frac{29}{12}x=-\frac{14}{12}
Dividing by 12 undoes the multiplication by 12.
x^{2}+\frac{29}{12}x=-\frac{7}{6}
Reduce the fraction \frac{-14}{12} to lowest terms by extracting and canceling out 2.
x^{2}+\frac{29}{12}x+\left(\frac{29}{24}\right)^{2}=-\frac{7}{6}+\left(\frac{29}{24}\right)^{2}
Divide \frac{29}{12}, the coefficient of the x term, by 2 to get \frac{29}{24}. Then add the square of \frac{29}{24} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{29}{12}x+\frac{841}{576}=-\frac{7}{6}+\frac{841}{576}
Square \frac{29}{24} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{29}{12}x+\frac{841}{576}=\frac{169}{576}
Add -\frac{7}{6} to \frac{841}{576} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{29}{24}\right)^{2}=\frac{169}{576}
Factor x^{2}+\frac{29}{12}x+\frac{841}{576}. 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{29}{24}\right)^{2}}=\sqrt{\frac{169}{576}}
Take the square root of both sides of the equation.
x+\frac{29}{24}=\frac{13}{24} x+\frac{29}{24}=-\frac{13}{24}
Simplify.
x=-\frac{2}{3} x=-\frac{7}{4}
Subtract \frac{29}{24} from both sides of the equation.
x ^ 2 +\frac{29}{12}x +\frac{7}{6} = 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 12
r + s = -\frac{29}{12} rs = \frac{7}{6}
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{29}{24} - u s = -\frac{29}{24} + u
Two numbers r and s sum up to -\frac{29}{12} exactly when the average of the two numbers is \frac{1}{2}*-\frac{29}{12} = -\frac{29}{24}. 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{29}{24} - u) (-\frac{29}{24} + u) = \frac{7}{6}
To solve for unknown quantity u, substitute these in the product equation rs = \frac{7}{6}
\frac{841}{576} - u^2 = \frac{7}{6}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = \frac{7}{6}-\frac{841}{576} = -\frac{169}{576}
Simplify the expression by subtracting \frac{841}{576} on both sides
u^2 = \frac{169}{576} u = \pm\sqrt{\frac{169}{576}} = \pm \frac{13}{24}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-\frac{29}{24} - \frac{13}{24} = -1.750 s = -\frac{29}{24} + \frac{13}{24} = -0.667
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.