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a+b=-4 ab=4
To solve the equation, factor b^{2}-4b+4 using formula b^{2}+\left(a+b\right)b+ab=\left(b+a\right)\left(b+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(b-2\right)\left(b-2\right)
Rewrite factored expression \left(b+a\right)\left(b+b\right) using the obtained values.
\left(b-2\right)^{2}
Rewrite as a binomial square.
b=2
To find equation solution, solve b-2=0.
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 b^{2}+ab+bb+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(b^{2}-2b\right)+\left(-2b+4\right)
Rewrite b^{2}-4b+4 as \left(b^{2}-2b\right)+\left(-2b+4\right).
b\left(b-2\right)-2\left(b-2\right)
Factor out b in the first and -2 in the second group.
\left(b-2\right)\left(b-2\right)
Factor out common term b-2 by using distributive property.
\left(b-2\right)^{2}
Rewrite as a binomial square.
b=2
To find equation solution, solve b-2=0.
b^{2}-4b+4=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.
b=\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}.
b=\frac{-\left(-4\right)±\sqrt{16-4\times 4}}{2}
Square -4.
b=\frac{-\left(-4\right)±\sqrt{16-16}}{2}
Multiply -4 times 4.
b=\frac{-\left(-4\right)±\sqrt{0}}{2}
Add 16 to -16.
b=-\frac{-4}{2}
Take the square root of 0.
b=\frac{4}{2}
The opposite of -4 is 4.
b=2
Divide 4 by 2.
b^{2}-4b+4=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.
\left(b-2\right)^{2}=0
Factor b^{2}-4b+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(b-2\right)^{2}}=\sqrt{0}
Take the square root of both sides of the equation.
b-2=0 b-2=0
Simplify.
b=2 b=2
Add 2 to both sides of the equation.
b=2
The equation is now solved. Solutions are the same.
x ^ 2 -4x +4 = 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.
r + s = 4 rs = 4
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 = 2 - u s = 2 + u
Two numbers r and s sum up to 4 exactly when the average of the two numbers is \frac{1}{2}*4 = 2. 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>
(2 - u) (2 + u) = 4
To solve for unknown quantity u, substitute these in the product equation rs = 4
4 - u^2 = 4
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 4-4 = 0
Simplify the expression by subtracting 4 on both sides
u^2 = 0 u = 0
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r = s = 2
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.