Solve for x
x=5
x=7
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x^{2}-12x+31+4=0
Add 4 to both sides.
x^{2}-12x+35=0
Add 31 and 4 to get 35.
a+b=-12 ab=35
To solve the equation, factor x^{2}-12x+35 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,-35 -5,-7
Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 35.
-1-35=-36 -5-7=-12
Calculate the sum for each pair.
a=-7 b=-5
The solution is the pair that gives sum -12.
\left(x-7\right)\left(x-5\right)
Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.
x=7 x=5
To find equation solutions, solve x-7=0 and x-5=0.
x^{2}-12x+31+4=0
Add 4 to both sides.
x^{2}-12x+35=0
Add 31 and 4 to get 35.
a+b=-12 ab=1\times 35=35
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+35. To find a and b, set up a system to be solved.
-1,-35 -5,-7
Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 35.
-1-35=-36 -5-7=-12
Calculate the sum for each pair.
a=-7 b=-5
The solution is the pair that gives sum -12.
\left(x^{2}-7x\right)+\left(-5x+35\right)
Rewrite x^{2}-12x+35 as \left(x^{2}-7x\right)+\left(-5x+35\right).
x\left(x-7\right)-5\left(x-7\right)
Factor out x in the first and -5 in the second group.
\left(x-7\right)\left(x-5\right)
Factor out common term x-7 by using distributive property.
x=7 x=5
To find equation solutions, solve x-7=0 and x-5=0.
x^{2}-12x+31=-4
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^{2}-12x+31-\left(-4\right)=-4-\left(-4\right)
Add 4 to both sides of the equation.
x^{2}-12x+31-\left(-4\right)=0
Subtracting -4 from itself leaves 0.
x^{2}-12x+35=0
Subtract -4 from 31.
x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 35}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -12 for b, and 35 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-12\right)±\sqrt{144-4\times 35}}{2}
Square -12.
x=\frac{-\left(-12\right)±\sqrt{144-140}}{2}
Multiply -4 times 35.
x=\frac{-\left(-12\right)±\sqrt{4}}{2}
Add 144 to -140.
x=\frac{-\left(-12\right)±2}{2}
Take the square root of 4.
x=\frac{12±2}{2}
The opposite of -12 is 12.
x=\frac{14}{2}
Now solve the equation x=\frac{12±2}{2} when ± is plus. Add 12 to 2.
x=7
Divide 14 by 2.
x=\frac{10}{2}
Now solve the equation x=\frac{12±2}{2} when ± is minus. Subtract 2 from 12.
x=5
Divide 10 by 2.
x=7 x=5
The equation is now solved.
x^{2}-12x+31=-4
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.
x^{2}-12x+31-31=-4-31
Subtract 31 from both sides of the equation.
x^{2}-12x=-4-31
Subtracting 31 from itself leaves 0.
x^{2}-12x=-35
Subtract 31 from -4.
x^{2}-12x+\left(-6\right)^{2}=-35+\left(-6\right)^{2}
Divide -12, the coefficient of the x term, by 2 to get -6. Then add the square of -6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-12x+36=-35+36
Square -6.
x^{2}-12x+36=1
Add -35 to 36.
\left(x-6\right)^{2}=1
Factor x^{2}-12x+36. 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-6\right)^{2}}=\sqrt{1}
Take the square root of both sides of the equation.
x-6=1 x-6=-1
Simplify.
x=7 x=5
Add 6 to both sides of the equation.
Examples
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{ 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}