Solve m^2-47m+400

$$m ^ { 2 } - 47 m + 400$$

Factor

$$\left(m-\frac{47-\sqrt{609}}{2}\right)\left(m-\frac{\sqrt{609}+47}{2}\right)$$

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.

$$m^{2}-47m+400=0$$

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.

$$m=\frac{-\left(-47\right)±\sqrt{\left(-47\right)^{2}-4\times 400}}{2}$$

Square -47.

$$m=\frac{-\left(-47\right)±\sqrt{2209-4\times 400}}{2}$$

Multiply -4 times 400.

$$m=\frac{-\left(-47\right)±\sqrt{2209-1600}}{2}$$

Add 2209 to -1600.

$$m=\frac{-\left(-47\right)±\sqrt{609}}{2}$$

The opposite of -47 is 47.

$$m=\frac{47±\sqrt{609}}{2}$$

Now solve the equation m=\frac{47±\sqrt{609}}{2} when ± is plus. Add 47 to \sqrt{609}.

$$m=\frac{\sqrt{609}+47}{2}$$

Now solve the equation m=\frac{47±\sqrt{609}}{2} when ± is minus. Subtract \sqrt{609} from 47.

$$m=\frac{47-\sqrt{609}}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{47+\sqrt{609}}{2} for x_{1} and \frac{47-\sqrt{609}}{2} for x_{2}.

$$m^{2}-47m+400=\left(m-\frac{\sqrt{609}+47}{2}\right)\left(m-\frac{47-\sqrt{609}}{2}\right)$$

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.

$$x ^ 2 -47x +400 = 0$$

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

$$r + s = 47 rs = 400$$

Two numbers r and s sum up to 47 exactly when the average of the two numbers is \frac{1}{2}*47 = \frac{47}{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>

$$r = \frac{47}{2} - u s = \frac{47}{2} + u$$

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

$$(\frac{47}{2} - u) (\frac{47}{2} + u) = 400$$

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

$$\frac{2209}{4} - u^2 = 400$$

Simplify the expression by subtracting \frac{2209}{4} on both sides

$$-u^2 = 400-\frac{2209}{4} = -\frac{609}{4}$$

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

$$u^2 = \frac{609}{4} u = \pm\sqrt{\frac{609}{4}} = \pm \frac{\sqrt{609}}{2}$$

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

$$r =\frac{47}{2} - \frac{\sqrt{609}}{2} = 11.161 s = \frac{47}{2} + \frac{\sqrt{609}}{2} = 35.839$$

Evaluate

$$m^{2}-47m+400$$