Solve p(x)=4x^2-52x+1

$$p ( x ) = 4 x ^ { 2 } - 52 x + 1$$

Factor

$$4\left(x-\left(\frac{13}{2}-\sqrt{42}\right)\right)\left(x-\left(\sqrt{42}+\frac{13}{2}\right)\right)$$

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.

$$4x^{2}-52x+1=0$$

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.

$$x=\frac{-\left(-52\right)±\sqrt{\left(-52\right)^{2}-4\times 4}}{2\times 4}$$

Square -52.

$$x=\frac{-\left(-52\right)±\sqrt{2704-4\times 4}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-52\right)±\sqrt{2704-16}}{2\times 4}$$

Add 2704 to -16.

$$x=\frac{-\left(-52\right)±\sqrt{2688}}{2\times 4}$$

Take the square root of 2688.

$$x=\frac{-\left(-52\right)±8\sqrt{42}}{2\times 4}$$

The opposite of -52 is 52.

$$x=\frac{52±8\sqrt{42}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{52±8\sqrt{42}}{8}$$

Now solve the equation x=\frac{52±8\sqrt{42}}{8} when ± is plus. Add 52 to 8\sqrt{42}.

$$x=\frac{8\sqrt{42}+52}{8}$$

Divide 52+8\sqrt{42} by 8.

$$x=\sqrt{42}+\frac{13}{2}$$

Now solve the equation x=\frac{52±8\sqrt{42}}{8} when ± is minus. Subtract 8\sqrt{42} from 52.

$$x=\frac{52-8\sqrt{42}}{8}$$

Divide 52-8\sqrt{42} by 8.

$$x=\frac{13}{2}-\sqrt{42}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{13}{2}+\sqrt{42} for x_{1} and \frac{13}{2}-\sqrt{42} for x_{2}.

$$4x^{2}-52x+1=4\left(x-\left(\sqrt{42}+\frac{13}{2}\right)\right)\left(x-\left(\frac{13}{2}-\sqrt{42}\right)\right)$$

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.This is achieved by dividing both sides of the equation by 4

$$x ^ 2 -13x +\frac{1}{4} = 0$$

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

$$r + s = 13 rs = \frac{1}{4}$$

Two numbers r and s sum up to 13 exactly when the average of the two numbers is \frac{1}{2}*13 = \frac{13}{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>

$$r = \frac{13}{2} - u s = \frac{13}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{1}{4}

$$(\frac{13}{2} - u) (\frac{13}{2} + u) = \frac{1}{4}$$

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

$$\frac{169}{4} - u^2 = \frac{1}{4}$$

Simplify the expression by subtracting \frac{169}{4} on both sides

$$-u^2 = \frac{1}{4}-\frac{169}{4} = -42$$

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

$$u^2 = 42 u = \pm\sqrt{42} = \pm \sqrt{42}$$

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

$$r =\frac{13}{2} - \sqrt{42} = 0.019 s = \frac{13}{2} + \sqrt{42} = 12.981$$

Evaluate

$$4x^{2}-52x+1$$