Solve 54x^2+863x-432

$$54 x ^ { 2 } + 863 x - 432$$

Factor

$$54\left(x-\frac{-\sqrt{838081}-863}{108}\right)\left(x-\frac{\sqrt{838081}-863}{108}\right)$$

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.

$$54x^{2}+863x-432=0$$

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.

$$x=\frac{-863±\sqrt{863^{2}-4\times 54\left(-432\right)}}{2\times 54}$$

Square 863.

$$x=\frac{-863±\sqrt{744769-4\times 54\left(-432\right)}}{2\times 54}$$

Multiply -4 times 54.

$$x=\frac{-863±\sqrt{744769-216\left(-432\right)}}{2\times 54}$$

Multiply -216 times -432.

$$x=\frac{-863±\sqrt{744769+93312}}{2\times 54}$$

Add 744769 to 93312.

$$x=\frac{-863±\sqrt{838081}}{2\times 54}$$

Multiply 2 times 54.

$$x=\frac{-863±\sqrt{838081}}{108}$$

Now solve the equation x=\frac{-863±\sqrt{838081}}{108} when ± is plus. Add -863 to \sqrt{838081}.

$$x=\frac{\sqrt{838081}-863}{108}$$

Now solve the equation x=\frac{-863±\sqrt{838081}}{108} when ± is minus. Subtract \sqrt{838081} from -863.

$$x=\frac{-\sqrt{838081}-863}{108}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-863+\sqrt{838081}}{108} for x_{1} and \frac{-863-\sqrt{838081}}{108} for x_{2}.

$$54x^{2}+863x-432=54\left(x-\frac{\sqrt{838081}-863}{108}\right)\left(x-\frac{-\sqrt{838081}-863}{108}\right)$$

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 54

$$x ^ 2 +\frac{863}{54}x -8 = 0$$

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

$$r + s = -\frac{863}{54} rs = -8$$

Two numbers r and s sum up to -\frac{863}{54} exactly when the average of the two numbers is \frac{1}{2}*-\frac{863}{54} = -\frac{863}{108}. 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>

$$r = -\frac{863}{108} - u s = -\frac{863}{108} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -8

$$(-\frac{863}{108} - u) (-\frac{863}{108} + u) = -8$$

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

$$\frac{744769}{11664} - u^2 = -8$$

Simplify the expression by subtracting \frac{744769}{11664} on both sides

$$-u^2 = -8-\frac{744769}{11664} = -\frac{838081}{11664}$$

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

$$u^2 = \frac{838081}{11664} u = \pm\sqrt{\frac{838081}{11664}} = \pm \frac{\sqrt{838081}}{108}$$

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

$$r =-\frac{863}{108} - \frac{\sqrt{838081}}{108} = -16.467 s = -\frac{863}{108} + \frac{\sqrt{838081}}{108} = 0.486$$

Evaluate

$$54x^{2}+863x-432$$