Solve for x
x=2
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x^{2}-4x+5-1=0
Subtract 1 from both sides.
x^{2}-4x+4=0
Subtract 1 from 5 to get 4.
a+b=-4 ab=4
To solve the equation, factor x^{2}-4x+4 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,-4 -2,-2
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 4.
-1-4=-5 -2-2=-4
Calculate the sum for each pair.
a=-2 b=-2
The solution is the pair that gives sum -4.
\left(x-2\right)\left(x-2\right)
Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.
\left(x-2\right)^{2}
Rewrite as a binomial square.
x=2
To find equation solution, solve x-2=0.
x^{2}-4x+5-1=0
Subtract 1 from both sides.
x^{2}-4x+4=0
Subtract 1 from 5 to get 4.
a+b=-4 ab=1\times 4=4
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+4. To find a and b, set up a system to be solved.
-1,-4 -2,-2
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 4.
-1-4=-5 -2-2=-4
Calculate the sum for each pair.
a=-2 b=-2
The solution is the pair that gives sum -4.
\left(x^{2}-2x\right)+\left(-2x+4\right)
Rewrite x^{2}-4x+4 as \left(x^{2}-2x\right)+\left(-2x+4\right).
x\left(x-2\right)-2\left(x-2\right)
Factor out x in the first and -2 in the second group.
\left(x-2\right)\left(x-2\right)
Factor out common term x-2 by using distributive property.
\left(x-2\right)^{2}
Rewrite as a binomial square.
x=2
To find equation solution, solve x-2=0.
x^{2}-4x+5=1
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}-4x+5-1=1-1
Subtract 1 from both sides of the equation.
x^{2}-4x+5-1=0
Subtracting 1 from itself leaves 0.
x^{2}-4x+4=0
Subtract 1 from 5.
x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 4}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -4 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-4\right)±\sqrt{16-4\times 4}}{2}
Square -4.
x=\frac{-\left(-4\right)±\sqrt{16-16}}{2}
Multiply -4 times 4.
x=\frac{-\left(-4\right)±\sqrt{0}}{2}
Add 16 to -16.
x=-\frac{-4}{2}
Take the square root of 0.
x=\frac{4}{2}
The opposite of -4 is 4.
x=2
Divide 4 by 2.
x^{2}-4x+5=1
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}-4x+5-5=1-5
Subtract 5 from both sides of the equation.
x^{2}-4x=1-5
Subtracting 5 from itself leaves 0.
x^{2}-4x=-4
Subtract 5 from 1.
x^{2}-4x+\left(-2\right)^{2}=-4+\left(-2\right)^{2}
Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-4x+4=-4+4
Square -2.
x^{2}-4x+4=0
Add -4 to 4.
\left(x-2\right)^{2}=0
Factor x^{2}-4x+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-2\right)^{2}}=\sqrt{0}
Take the square root of both sides of the equation.
x-2=0 x-2=0
Simplify.
x=2 x=2
Add 2 to both sides of the equation.
x=2
The equation is now solved. Solutions are the same.
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}