Solve for x
x = -\frac{7}{4} = -1\frac{3}{4} = -1.75
x=\frac{2}{3}\approx 0.666666667
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a+b=13 ab=12\left(-14\right)=-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 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 -168.
-1+168=167 -2+84=82 -3+56=53 -4+42=38 -6+28=22 -7+24=17 -8+21=13 -12+14=2
Calculate the sum for each pair.
a=-8 b=21
The solution is the pair that gives sum 13.
\left(12x^{2}-8x\right)+\left(21x-14\right)
Rewrite 12x^{2}+13x-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}+13x-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{-13±\sqrt{13^{2}-4\times 12\left(-14\right)}}{2\times 12}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 12 for a, 13 for b, and -14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-13±\sqrt{169-4\times 12\left(-14\right)}}{2\times 12}
Square 13.
x=\frac{-13±\sqrt{169-48\left(-14\right)}}{2\times 12}
Multiply -4 times 12.
x=\frac{-13±\sqrt{169+672}}{2\times 12}
Multiply -48 times -14.
x=\frac{-13±\sqrt{841}}{2\times 12}
Add 169 to 672.
x=\frac{-13±29}{2\times 12}
Take the square root of 841.
x=\frac{-13±29}{24}
Multiply 2 times 12.
x=\frac{16}{24}
Now solve the equation x=\frac{-13±29}{24} when ± is plus. Add -13 to 29.
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{-13±29}{24} when ± is minus. Subtract 29 from -13.
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}+13x-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}+13x-14-\left(-14\right)=-\left(-14\right)
Add 14 to both sides of the equation.
12x^{2}+13x=-\left(-14\right)
Subtracting -14 from itself leaves 0.
12x^{2}+13x=14
Subtract -14 from 0.
\frac{12x^{2}+13x}{12}=\frac{14}{12}
Divide both sides by 12.
x^{2}+\frac{13}{12}x=\frac{14}{12}
Dividing by 12 undoes the multiplication by 12.
x^{2}+\frac{13}{12}x=\frac{7}{6}
Reduce the fraction \frac{14}{12} to lowest terms by extracting and canceling out 2.
x^{2}+\frac{13}{12}x+\left(\frac{13}{24}\right)^{2}=\frac{7}{6}+\left(\frac{13}{24}\right)^{2}
Divide \frac{13}{12}, the coefficient of the x term, by 2 to get \frac{13}{24}. Then add the square of \frac{13}{24} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{13}{12}x+\frac{169}{576}=\frac{7}{6}+\frac{169}{576}
Square \frac{13}{24} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{13}{12}x+\frac{169}{576}=\frac{841}{576}
Add \frac{7}{6} to \frac{169}{576} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{13}{24}\right)^{2}=\frac{841}{576}
Factor x^{2}+\frac{13}{12}x+\frac{169}{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{13}{24}\right)^{2}}=\sqrt{\frac{841}{576}}
Take the square root of both sides of the equation.
x+\frac{13}{24}=\frac{29}{24} x+\frac{13}{24}=-\frac{29}{24}
Simplify.
x=\frac{2}{3} x=-\frac{7}{4}
Subtract \frac{13}{24} from both sides of the equation.
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Linear equation
y = 3x + 4
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Matrix
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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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