Solve for x
x=16
x=25
Graph
Share
Copied to clipboard
a+b=-41 ab=400
To solve the equation, factor x^{2}-41x+400 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,-400 -2,-200 -4,-100 -5,-80 -8,-50 -10,-40 -16,-25 -20,-20
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 400.
-1-400=-401 -2-200=-202 -4-100=-104 -5-80=-85 -8-50=-58 -10-40=-50 -16-25=-41 -20-20=-40
Calculate the sum for each pair.
a=-25 b=-16
The solution is the pair that gives sum -41.
\left(x-25\right)\left(x-16\right)
Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.
x=25 x=16
To find equation solutions, solve x-25=0 and x-16=0.
a+b=-41 ab=1\times 400=400
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+400. To find a and b, set up a system to be solved.
-1,-400 -2,-200 -4,-100 -5,-80 -8,-50 -10,-40 -16,-25 -20,-20
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 400.
-1-400=-401 -2-200=-202 -4-100=-104 -5-80=-85 -8-50=-58 -10-40=-50 -16-25=-41 -20-20=-40
Calculate the sum for each pair.
a=-25 b=-16
The solution is the pair that gives sum -41.
\left(x^{2}-25x\right)+\left(-16x+400\right)
Rewrite x^{2}-41x+400 as \left(x^{2}-25x\right)+\left(-16x+400\right).
x\left(x-25\right)-16\left(x-25\right)
Factor out x in the first and -16 in the second group.
\left(x-25\right)\left(x-16\right)
Factor out common term x-25 by using distributive property.
x=25 x=16
To find equation solutions, solve x-25=0 and x-16=0.
x^{2}-41x+400=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{-\left(-41\right)±\sqrt{\left(-41\right)^{2}-4\times 400}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -41 for b, and 400 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-41\right)±\sqrt{1681-4\times 400}}{2}
Square -41.
x=\frac{-\left(-41\right)±\sqrt{1681-1600}}{2}
Multiply -4 times 400.
x=\frac{-\left(-41\right)±\sqrt{81}}{2}
Add 1681 to -1600.
x=\frac{-\left(-41\right)±9}{2}
Take the square root of 81.
x=\frac{41±9}{2}
The opposite of -41 is 41.
x=\frac{50}{2}
Now solve the equation x=\frac{41±9}{2} when ± is plus. Add 41 to 9.
x=25
Divide 50 by 2.
x=\frac{32}{2}
Now solve the equation x=\frac{41±9}{2} when ± is minus. Subtract 9 from 41.
x=16
Divide 32 by 2.
x=25 x=16
The equation is now solved.
x^{2}-41x+400=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.
x^{2}-41x+400-400=-400
Subtract 400 from both sides of the equation.
x^{2}-41x=-400
Subtracting 400 from itself leaves 0.
x^{2}-41x+\left(-\frac{41}{2}\right)^{2}=-400+\left(-\frac{41}{2}\right)^{2}
Divide -41, the coefficient of the x term, by 2 to get -\frac{41}{2}. Then add the square of -\frac{41}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-41x+\frac{1681}{4}=-400+\frac{1681}{4}
Square -\frac{41}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}-41x+\frac{1681}{4}=\frac{81}{4}
Add -400 to \frac{1681}{4}.
\left(x-\frac{41}{2}\right)^{2}=\frac{81}{4}
Factor x^{2}-41x+\frac{1681}{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{41}{2}\right)^{2}}=\sqrt{\frac{81}{4}}
Take the square root of both sides of the equation.
x-\frac{41}{2}=\frac{9}{2} x-\frac{41}{2}=-\frac{9}{2}
Simplify.
x=25 x=16
Add \frac{41}{2} to both sides of the equation.
x ^ 2 -41x +400 = 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 = 41 rs = 400
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 = \frac{41}{2} - u s = \frac{41}{2} + u
Two numbers r and s sum up to 41 exactly when the average of the two numbers is \frac{1}{2}*41 = \frac{41}{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>
(\frac{41}{2} - u) (\frac{41}{2} + u) = 400
To solve for unknown quantity u, substitute these in the product equation rs = 400
\frac{1681}{4} - u^2 = 400
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 400-\frac{1681}{4} = -\frac{81}{4}
Simplify the expression by subtracting \frac{1681}{4} on both sides
u^2 = \frac{81}{4} u = \pm\sqrt{\frac{81}{4}} = \pm \frac{9}{2}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =\frac{41}{2} - \frac{9}{2} = 16 s = \frac{41}{2} + \frac{9}{2} = 25
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
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}