Solve -117x^2+477x+36

$$- 117 x ^ { 2 } + 477 x + 36$$

Factor

$$-117\left(x-\frac{53-\sqrt{3017}}{26}\right)\left(x-\frac{\sqrt{3017}+53}{26}\right)$$

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.

$$-117x^{2}+477x+36=0$$

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.

$$x=\frac{-477±\sqrt{477^{2}-4\left(-117\right)\times 36}}{2\left(-117\right)}$$

Square 477.

$$x=\frac{-477±\sqrt{227529-4\left(-117\right)\times 36}}{2\left(-117\right)}$$

Multiply -4 times -117.

$$x=\frac{-477±\sqrt{227529+468\times 36}}{2\left(-117\right)}$$

Multiply 468 times 36.

$$x=\frac{-477±\sqrt{227529+16848}}{2\left(-117\right)}$$

Add 227529 to 16848.

$$x=\frac{-477±\sqrt{244377}}{2\left(-117\right)}$$

Take the square root of 244377.

$$x=\frac{-477±9\sqrt{3017}}{2\left(-117\right)}$$

Multiply 2 times -117.

$$x=\frac{-477±9\sqrt{3017}}{-234}$$

Now solve the equation x=\frac{-477±9\sqrt{3017}}{-234} when ± is plus. Add -477 to 9\sqrt{3017}.

$$x=\frac{9\sqrt{3017}-477}{-234}$$

Divide -477+9\sqrt{3017} by -234.

$$x=\frac{53-\sqrt{3017}}{26}$$

Now solve the equation x=\frac{-477±9\sqrt{3017}}{-234} when ± is minus. Subtract 9\sqrt{3017} from -477.

$$x=\frac{-9\sqrt{3017}-477}{-234}$$

Divide -477-9\sqrt{3017} by -234.

$$x=\frac{\sqrt{3017}+53}{26}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{53-\sqrt{3017}}{26} for x_{1} and \frac{53+\sqrt{3017}}{26} for x_{2}.

$$-117x^{2}+477x+36=-117\left(x-\frac{53-\sqrt{3017}}{26}\right)\left(x-\frac{\sqrt{3017}+53}{26}\right)$$

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.

$$x ^ 2 -\frac{53}{13}x -\frac{4}{13} = 0$$

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

$$r + s = \frac{53}{13} rs = -\frac{4}{13}$$

Two numbers r and s sum up to \frac{53}{13} exactly when the average of the two numbers is \frac{1}{2}*\frac{53}{13} = \frac{53}{26}. 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>

$$r = \frac{53}{26} - u s = \frac{53}{26} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{4}{13}

$$(\frac{53}{26} - u) (\frac{53}{26} + u) = -\frac{4}{13}$$

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

$$\frac{2809}{676} - u^2 = -\frac{4}{13}$$

Simplify the expression by subtracting \frac{2809}{676} on both sides

$$-u^2 = -\frac{4}{13}-\frac{2809}{676} = -\frac{3017}{676}$$

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

$$u^2 = \frac{3017}{676} u = \pm\sqrt{\frac{3017}{676}} = \pm \frac{\sqrt{3017}}{26}$$

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

$$r =\frac{53}{26} - \frac{\sqrt{3017}}{26} = -0.074 s = \frac{53}{26} + \frac{\sqrt{3017}}{26} = 4.151$$

Evaluate

$$36+477x-117x^{2}$$