Solve 75x^2-766x+80

$$75 x ^ { 2 } - 766 x + 80$$

Factor

$$75\left(x-\frac{383-\sqrt{140689}}{75}\right)\left(x-\frac{\sqrt{140689}+383}{75}\right)$$

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.

$$75x^{2}-766x+80=0$$

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.

$$x=\frac{-\left(-766\right)±\sqrt{\left(-766\right)^{2}-4\times 75\times 80}}{2\times 75}$$

Square -766.

$$x=\frac{-\left(-766\right)±\sqrt{586756-4\times 75\times 80}}{2\times 75}$$

Multiply -4 times 75.

$$x=\frac{-\left(-766\right)±\sqrt{586756-300\times 80}}{2\times 75}$$

Multiply -300 times 80.

$$x=\frac{-\left(-766\right)±\sqrt{586756-24000}}{2\times 75}$$

Add 586756 to -24000.

$$x=\frac{-\left(-766\right)±\sqrt{562756}}{2\times 75}$$

Take the square root of 562756.

$$x=\frac{-\left(-766\right)±2\sqrt{140689}}{2\times 75}$$

The opposite of -766 is 766.

$$x=\frac{766±2\sqrt{140689}}{2\times 75}$$

Multiply 2 times 75.

$$x=\frac{766±2\sqrt{140689}}{150}$$

Now solve the equation x=\frac{766±2\sqrt{140689}}{150} when ± is plus. Add 766 to 2\sqrt{140689}.

$$x=\frac{2\sqrt{140689}+766}{150}$$

Divide 766+2\sqrt{140689} by 150.

$$x=\frac{\sqrt{140689}+383}{75}$$

Now solve the equation x=\frac{766±2\sqrt{140689}}{150} when ± is minus. Subtract 2\sqrt{140689} from 766.

$$x=\frac{766-2\sqrt{140689}}{150}$$

Divide 766-2\sqrt{140689} by 150.

$$x=\frac{383-\sqrt{140689}}{75}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{383+\sqrt{140689}}{75} for x_{1} and \frac{383-\sqrt{140689}}{75} for x_{2}.

$$75x^{2}-766x+80=75\left(x-\frac{\sqrt{140689}+383}{75}\right)\left(x-\frac{383-\sqrt{140689}}{75}\right)$$

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 75

$$x ^ 2 -\frac{766}{75}x +\frac{16}{15} = 0$$

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

$$r + s = \frac{766}{75} rs = \frac{16}{15}$$

Two numbers r and s sum up to \frac{766}{75} exactly when the average of the two numbers is \frac{1}{2}*\frac{766}{75} = \frac{383}{75}. 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>

$$r = \frac{383}{75} - u s = \frac{383}{75} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{16}{15}

$$(\frac{383}{75} - u) (\frac{383}{75} + u) = \frac{16}{15}$$

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

$$\frac{146689}{5625} - u^2 = \frac{16}{15}$$

Simplify the expression by subtracting \frac{146689}{5625} on both sides

$$-u^2 = \frac{16}{15}-\frac{146689}{5625} = -\frac{140689}{5625}$$

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

$$u^2 = \frac{140689}{5625} u = \pm\sqrt{\frac{140689}{5625}} = \pm \frac{\sqrt{140689}}{75}$$

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

$$r =\frac{383}{75} - \frac{\sqrt{140689}}{75} = 0.106 s = \frac{383}{75} + \frac{\sqrt{140689}}{75} = 10.108$$

Evaluate

$$75x^{2}-766x+80$$