Solve f^2-41f+400

$$f ^ { 2 } - 41 f + 400$$

Factor

$$\left(f-25\right)\left(f-16\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as f^{2}+af+bf+400. To find a and b, set up a system to be solved.

$$a+b=-41 ab=1\times 400=400$$

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.

$$-1,-400 -2,-200 -4,-100 -5,-80 -8,-50 -10,-40 -16,-25 -20,-20$$

Calculate the sum for each pair.

$$-1-400=-401 -2-200=-202 -4-100=-104 -5-80=-85 -8-50=-58 -10-40=-50 -16-25=-41 -20-20=-40$$

The solution is the pair that gives sum -41.

$$a=-25 b=-16$$

Rewrite f^{2}-41f+400 as \left(f^{2}-25f\right)+\left(-16f+400\right).

$$\left(f^{2}-25f\right)+\left(-16f+400\right)$$

Factor out f in the first and -16 in the second group.

$$f\left(f-25\right)-16\left(f-25\right)$$

Factor out common term f-25 by using distributive property.

$$\left(f-25\right)\left(f-16\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$f^{2}-41f+400=0$$

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.

$$f=\frac{-\left(-41\right)±\sqrt{\left(-41\right)^{2}-4\times 400}}{2}$$

Square -41.

$$f=\frac{-\left(-41\right)±\sqrt{1681-4\times 400}}{2}$$

Multiply -4 times 400.

$$f=\frac{-\left(-41\right)±\sqrt{1681-1600}}{2}$$

Add 1681 to -1600.

$$f=\frac{-\left(-41\right)±\sqrt{81}}{2}$$

Take the square root of 81.

$$f=\frac{-\left(-41\right)±9}{2}$$

The opposite of -41 is 41.

$$f=\frac{41±9}{2}$$

Now solve the equation f=\frac{41±9}{2} when ± is plus. Add 41 to 9.

$$f=\frac{50}{2}$$

Divide 50 by 2.

$$f=25$$

Now solve the equation f=\frac{41±9}{2} when ± is minus. Subtract 9 from 41.

$$f=\frac{32}{2}$$

Divide 32 by 2.

$$f=16$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 25 for x_{1} and 16 for x_{2}.

$$f^{2}-41f+400=\left(f-25\right)\left(f-16\right)$$

Steps Using Direct Factoring Method

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.

$$x ^ 2 -41x +400 = 0$$

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

$$r + s = 41 rs = 400$$

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>

$$r = \frac{41}{2} - u s = \frac{41}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 400

$$(\frac{41}{2} - u) (\frac{41}{2} + u) = 400$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$\frac{1681}{4} - u^2 = 400$$

Simplify the expression by subtracting \frac{1681}{4} on both sides

$$-u^2 = 400-\frac{1681}{4} = -\frac{81}{4}$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = \frac{81}{4} u = \pm\sqrt{\frac{81}{4}} = \pm \frac{9}{2}$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =\frac{41}{2} - \frac{9}{2} = 16 s = \frac{41}{2} + \frac{9}{2} = 25$$

Evaluate

$$\left(f-25\right)\left(f-16\right)$$