Solve 4x^2-154x-720

$$4 x ^ { 2 } - 154 x - 720$$

Factor

$$4\left(x-\frac{77-\sqrt{8809}}{4}\right)\left(x-\frac{\sqrt{8809}+77}{4}\right)$$

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.

$$4x^{2}-154x-720=0$$

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.

$$x=\frac{-\left(-154\right)±\sqrt{\left(-154\right)^{2}-4\times 4\left(-720\right)}}{2\times 4}$$

Square -154.

$$x=\frac{-\left(-154\right)±\sqrt{23716-4\times 4\left(-720\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-154\right)±\sqrt{23716-16\left(-720\right)}}{2\times 4}$$

Multiply -16 times -720.

$$x=\frac{-\left(-154\right)±\sqrt{23716+11520}}{2\times 4}$$

Add 23716 to 11520.

$$x=\frac{-\left(-154\right)±\sqrt{35236}}{2\times 4}$$

Take the square root of 35236.

$$x=\frac{-\left(-154\right)±2\sqrt{8809}}{2\times 4}$$

The opposite of -154 is 154.

$$x=\frac{154±2\sqrt{8809}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{154±2\sqrt{8809}}{8}$$

Now solve the equation x=\frac{154±2\sqrt{8809}}{8} when ± is plus. Add 154 to 2\sqrt{8809}.

$$x=\frac{2\sqrt{8809}+154}{8}$$

Divide 154+2\sqrt{8809} by 8.

$$x=\frac{\sqrt{8809}+77}{4}$$

Now solve the equation x=\frac{154±2\sqrt{8809}}{8} when ± is minus. Subtract 2\sqrt{8809} from 154.

$$x=\frac{154-2\sqrt{8809}}{8}$$

Divide 154-2\sqrt{8809} by 8.

$$x=\frac{77-\sqrt{8809}}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{77+\sqrt{8809}}{4} for x_{1} and \frac{77-\sqrt{8809}}{4} for x_{2}.

$$4x^{2}-154x-720=4\left(x-\frac{\sqrt{8809}+77}{4}\right)\left(x-\frac{77-\sqrt{8809}}{4}\right)$$

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

$$x ^ 2 -\frac{77}{2}x -180 = 0$$

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

$$r + s = \frac{77}{2} rs = -180$$

Two numbers r and s sum up to \frac{77}{2} exactly when the average of the two numbers is \frac{1}{2}*\frac{77}{2} = \frac{77}{4}. 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>

$$r = \frac{77}{4} - u s = \frac{77}{4} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -180

$$(\frac{77}{4} - u) (\frac{77}{4} + u) = -180$$

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

$$\frac{5929}{16} - u^2 = -180$$

Simplify the expression by subtracting \frac{5929}{16} on both sides

$$-u^2 = -180-\frac{5929}{16} = -\frac{8809}{16}$$

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

$$u^2 = \frac{8809}{16} u = \pm\sqrt{\frac{8809}{16}} = \pm \frac{\sqrt{8809}}{4}$$

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

$$r =\frac{77}{4} - \frac{\sqrt{8809}}{4} = -4.214 s = \frac{77}{4} + \frac{\sqrt{8809}}{4} = 42.714$$

Evaluate

$$4x^{2}-154x-720$$