Solve for x
x=-4
x=-3
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x=\frac{-12}{x}-\frac{7x}{x}
To add or subtract expressions, expand them to make their denominators the same. Multiply 7 times \frac{x}{x}.
x=\frac{-12-7x}{x}
Since \frac{-12}{x} and \frac{7x}{x} have the same denominator, subtract them by subtracting their numerators.
x-\frac{-12-7x}{x}=0
Subtract \frac{-12-7x}{x} from both sides.
\frac{xx}{x}-\frac{-12-7x}{x}=0
To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x}{x}.
\frac{xx-\left(-12-7x\right)}{x}=0
Since \frac{xx}{x} and \frac{-12-7x}{x} have the same denominator, subtract them by subtracting their numerators.
\frac{x^{2}+12+7x}{x}=0
Do the multiplications in xx-\left(-12-7x\right).
x^{2}+12+7x=0
Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.
x^{2}+7x+12=0
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
a+b=7 ab=12
To solve the equation, factor x^{2}+7x+12 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.
1,12 2,6 3,4
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 12.
1+12=13 2+6=8 3+4=7
Calculate the sum for each pair.
a=3 b=4
The solution is the pair that gives sum 7.
\left(x+3\right)\left(x+4\right)
Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.
x=-3 x=-4
To find equation solutions, solve x+3=0 and x+4=0.
x=\frac{-12}{x}-\frac{7x}{x}
To add or subtract expressions, expand them to make their denominators the same. Multiply 7 times \frac{x}{x}.
x=\frac{-12-7x}{x}
Since \frac{-12}{x} and \frac{7x}{x} have the same denominator, subtract them by subtracting their numerators.
x-\frac{-12-7x}{x}=0
Subtract \frac{-12-7x}{x} from both sides.
\frac{xx}{x}-\frac{-12-7x}{x}=0
To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x}{x}.
\frac{xx-\left(-12-7x\right)}{x}=0
Since \frac{xx}{x} and \frac{-12-7x}{x} have the same denominator, subtract them by subtracting their numerators.
\frac{x^{2}+12+7x}{x}=0
Do the multiplications in xx-\left(-12-7x\right).
x^{2}+12+7x=0
Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.
x^{2}+7x+12=0
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
a+b=7 ab=1\times 12=12
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+12. To find a and b, set up a system to be solved.
1,12 2,6 3,4
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 12.
1+12=13 2+6=8 3+4=7
Calculate the sum for each pair.
a=3 b=4
The solution is the pair that gives sum 7.
\left(x^{2}+3x\right)+\left(4x+12\right)
Rewrite x^{2}+7x+12 as \left(x^{2}+3x\right)+\left(4x+12\right).
x\left(x+3\right)+4\left(x+3\right)
Factor out x in the first and 4 in the second group.
\left(x+3\right)\left(x+4\right)
Factor out common term x+3 by using distributive property.
x=-3 x=-4
To find equation solutions, solve x+3=0 and x+4=0.
x=\frac{-12}{x}-\frac{7x}{x}
To add or subtract expressions, expand them to make their denominators the same. Multiply 7 times \frac{x}{x}.
x=\frac{-12-7x}{x}
Since \frac{-12}{x} and \frac{7x}{x} have the same denominator, subtract them by subtracting their numerators.
x-\frac{-12-7x}{x}=0
Subtract \frac{-12-7x}{x} from both sides.
\frac{xx}{x}-\frac{-12-7x}{x}=0
To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x}{x}.
\frac{xx-\left(-12-7x\right)}{x}=0
Since \frac{xx}{x} and \frac{-12-7x}{x} have the same denominator, subtract them by subtracting their numerators.
\frac{x^{2}+12+7x}{x}=0
Do the multiplications in xx-\left(-12-7x\right).
x^{2}+12+7x=0
Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.
x^{2}+7x+12=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{-7±\sqrt{7^{2}-4\times 12}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 7 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-7±\sqrt{49-4\times 12}}{2}
Square 7.
x=\frac{-7±\sqrt{49-48}}{2}
Multiply -4 times 12.
x=\frac{-7±\sqrt{1}}{2}
Add 49 to -48.
x=\frac{-7±1}{2}
Take the square root of 1.
x=-\frac{6}{2}
Now solve the equation x=\frac{-7±1}{2} when ± is plus. Add -7 to 1.
x=-3
Divide -6 by 2.
x=-\frac{8}{2}
Now solve the equation x=\frac{-7±1}{2} when ± is minus. Subtract 1 from -7.
x=-4
Divide -8 by 2.
x=-3 x=-4
The equation is now solved.
x=\frac{-12}{x}-\frac{7x}{x}
To add or subtract expressions, expand them to make their denominators the same. Multiply 7 times \frac{x}{x}.
x=\frac{-12-7x}{x}
Since \frac{-12}{x} and \frac{7x}{x} have the same denominator, subtract them by subtracting their numerators.
x-\frac{-12-7x}{x}=0
Subtract \frac{-12-7x}{x} from both sides.
\frac{xx}{x}-\frac{-12-7x}{x}=0
To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x}{x}.
\frac{xx-\left(-12-7x\right)}{x}=0
Since \frac{xx}{x} and \frac{-12-7x}{x} have the same denominator, subtract them by subtracting their numerators.
\frac{x^{2}+12+7x}{x}=0
Do the multiplications in xx-\left(-12-7x\right).
x^{2}+12+7x=0
Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.
x^{2}+7x=-12
Subtract 12 from both sides. Anything subtracted from zero gives its negation.
x^{2}+7x+\left(\frac{7}{2}\right)^{2}=-12+\left(\frac{7}{2}\right)^{2}
Divide 7, the coefficient of the x term, by 2 to get \frac{7}{2}. Then add the square of \frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+7x+\frac{49}{4}=-12+\frac{49}{4}
Square \frac{7}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}+7x+\frac{49}{4}=\frac{1}{4}
Add -12 to \frac{49}{4}.
\left(x+\frac{7}{2}\right)^{2}=\frac{1}{4}
Factor x^{2}+7x+\frac{49}{4}. 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{7}{2}\right)^{2}}=\sqrt{\frac{1}{4}}
Take the square root of both sides of the equation.
x+\frac{7}{2}=\frac{1}{2} x+\frac{7}{2}=-\frac{1}{2}
Simplify.
x=-3 x=-4
Subtract \frac{7}{2} from both sides of the equation.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
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}