Solve x^2-38x+9

$$x ^ { 2 } - 38 x + 9$$

Factor

$$\left(x-\left(19-4\sqrt{22}\right)\right)\left(x-\left(4\sqrt{22}+19\right)\right)$$

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.

$$x^{2}-38x+9=0$$

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.

$$x=\frac{-\left(-38\right)±\sqrt{\left(-38\right)^{2}-4\times 9}}{2}$$

Square -38.

$$x=\frac{-\left(-38\right)±\sqrt{1444-4\times 9}}{2}$$

Multiply -4 times 9.

$$x=\frac{-\left(-38\right)±\sqrt{1444-36}}{2}$$

Add 1444 to -36.

$$x=\frac{-\left(-38\right)±\sqrt{1408}}{2}$$

Take the square root of 1408.

$$x=\frac{-\left(-38\right)±8\sqrt{22}}{2}$$

The opposite of -38 is 38.

$$x=\frac{38±8\sqrt{22}}{2}$$

Now solve the equation x=\frac{38±8\sqrt{22}}{2} when ± is plus. Add 38 to 8\sqrt{22}.

$$x=\frac{8\sqrt{22}+38}{2}$$

Divide 38+8\sqrt{22} by 2.

$$x=4\sqrt{22}+19$$

Now solve the equation x=\frac{38±8\sqrt{22}}{2} when ± is minus. Subtract 8\sqrt{22} from 38.

$$x=\frac{38-8\sqrt{22}}{2}$$

Divide 38-8\sqrt{22} by 2.

$$x=19-4\sqrt{22}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 19+4\sqrt{22} for x_{1} and 19-4\sqrt{22} for x_{2}.

$$x^{2}-38x+9=\left(x-\left(4\sqrt{22}+19\right)\right)\left(x-\left(19-4\sqrt{22}\right)\right)$$

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.

$$x ^ 2 -38x +9 = 0$$

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

$$r + s = 38 rs = 9$$

Two numbers r and s sum up to 38 exactly when the average of the two numbers is \frac{1}{2}*38 = 19. 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>

$$r = 19 - u s = 19 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 9

$$(19 - u) (19 + u) = 9$$

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

$$361 - u^2 = 9$$

Simplify the expression by subtracting 361 on both sides

$$-u^2 = 9-361 = -352$$

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

$$u^2 = 352 u = \pm\sqrt{352} = \pm \sqrt{352}$$

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

$$r =19 - \sqrt{352} = 0.238 s = 19 + \sqrt{352} = 37.762$$

Evaluate

$$x^{2}-38x+9$$