Solve x^2-156x-320

$$x ^ { 2 } - 156 x - 320$$

Factor

$$\left(x-\left(78-2\sqrt{1601}\right)\right)\left(x-\left(2\sqrt{1601}+78\right)\right)$$

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.

$$x^{2}-156x-320=0$$

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.

$$x=\frac{-\left(-156\right)±\sqrt{\left(-156\right)^{2}-4\left(-320\right)}}{2}$$

Square -156.

$$x=\frac{-\left(-156\right)±\sqrt{24336-4\left(-320\right)}}{2}$$

Multiply -4 times -320.

$$x=\frac{-\left(-156\right)±\sqrt{24336+1280}}{2}$$

Add 24336 to 1280.

$$x=\frac{-\left(-156\right)±\sqrt{25616}}{2}$$

Take the square root of 25616.

$$x=\frac{-\left(-156\right)±4\sqrt{1601}}{2}$$

The opposite of -156 is 156.

$$x=\frac{156±4\sqrt{1601}}{2}$$

Now solve the equation x=\frac{156±4\sqrt{1601}}{2} when ± is plus. Add 156 to 4\sqrt{1601}.

$$x=\frac{4\sqrt{1601}+156}{2}$$

Divide 156+4\sqrt{1601} by 2.

$$x=2\sqrt{1601}+78$$

Now solve the equation x=\frac{156±4\sqrt{1601}}{2} when ± is minus. Subtract 4\sqrt{1601} from 156.

$$x=\frac{156-4\sqrt{1601}}{2}$$

Divide 156-4\sqrt{1601} by 2.

$$x=78-2\sqrt{1601}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 78+2\sqrt{1601} for x_{1} and 78-2\sqrt{1601} for x_{2}.

$$x^{2}-156x-320=\left(x-\left(2\sqrt{1601}+78\right)\right)\left(x-\left(78-2\sqrt{1601}\right)\right)$$

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.

$$x ^ 2 -156x -320 = 0$$

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

$$r + s = 156 rs = -320$$

Two numbers r and s sum up to 156 exactly when the average of the two numbers is \frac{1}{2}*156 = 78. 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>

$$r = 78 - u s = 78 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -320

$$(78 - u) (78 + u) = -320$$

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

$$6084 - u^2 = -320$$

Simplify the expression by subtracting 6084 on both sides

$$-u^2 = -320-6084 = -6404$$

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

$$u^2 = 6404 u = \pm\sqrt{6404} = \pm \sqrt{6404}$$

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

$$r =78 - \sqrt{6404} = -2.025 s = 78 + \sqrt{6404} = 158.025$$

Evaluate

$$x^{2}-156x-320$$