Solve 150x^2-180x-57

$$150 x ^ { 2 } - 180 x - 57$$

Factor

$$150\left(x-\left(-\frac{\sqrt{74}}{10}+\frac{3}{5}\right)\right)\left(x-\left(\frac{\sqrt{74}}{10}+\frac{3}{5}\right)\right)$$

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.

$$150x^{2}-180x-57=0$$

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.

$$x=\frac{-\left(-180\right)±\sqrt{\left(-180\right)^{2}-4\times 150\left(-57\right)}}{2\times 150}$$

Square -180.

$$x=\frac{-\left(-180\right)±\sqrt{32400-4\times 150\left(-57\right)}}{2\times 150}$$

Multiply -4 times 150.

$$x=\frac{-\left(-180\right)±\sqrt{32400-600\left(-57\right)}}{2\times 150}$$

Multiply -600 times -57.

$$x=\frac{-\left(-180\right)±\sqrt{32400+34200}}{2\times 150}$$

Add 32400 to 34200.

$$x=\frac{-\left(-180\right)±\sqrt{66600}}{2\times 150}$$

Take the square root of 66600.

$$x=\frac{-\left(-180\right)±30\sqrt{74}}{2\times 150}$$

The opposite of -180 is 180.

$$x=\frac{180±30\sqrt{74}}{2\times 150}$$

Multiply 2 times 150.

$$x=\frac{180±30\sqrt{74}}{300}$$

Now solve the equation x=\frac{180±30\sqrt{74}}{300} when ± is plus. Add 180 to 30\sqrt{74}.

$$x=\frac{30\sqrt{74}+180}{300}$$

Divide 180+30\sqrt{74} by 300.

$$x=\frac{\sqrt{74}}{10}+\frac{3}{5}$$

Now solve the equation x=\frac{180±30\sqrt{74}}{300} when ± is minus. Subtract 30\sqrt{74} from 180.

$$x=\frac{180-30\sqrt{74}}{300}$$

Divide 180-30\sqrt{74} by 300.

$$x=-\frac{\sqrt{74}}{10}+\frac{3}{5}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{3}{5}+\frac{\sqrt{74}}{10} for x_{1} and \frac{3}{5}-\frac{\sqrt{74}}{10} for x_{2}.

$$150x^{2}-180x-57=150\left(x-\left(\frac{\sqrt{74}}{10}+\frac{3}{5}\right)\right)\left(x-\left(-\frac{\sqrt{74}}{10}+\frac{3}{5}\right)\right)$$

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.This is achieved by dividing both sides of the equation by 150

$$x ^ 2 -\frac{6}{5}x -\frac{19}{50} = 0$$

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

$$r + s = \frac{6}{5} rs = -\frac{19}{50}$$

Two numbers r and s sum up to \frac{6}{5} exactly when the average of the two numbers is \frac{1}{2}*\frac{6}{5} = \frac{3}{5}. 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>

$$r = \frac{3}{5} - u s = \frac{3}{5} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{19}{50}

$$(\frac{3}{5} - u) (\frac{3}{5} + u) = -\frac{19}{50}$$

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

$$\frac{9}{25} - u^2 = -\frac{19}{50}$$

Simplify the expression by subtracting \frac{9}{25} on both sides

$$-u^2 = -\frac{19}{50}-\frac{9}{25} = -\frac{37}{50}$$

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

$$u^2 = \frac{37}{50} u = \pm\sqrt{\frac{37}{50}} = \pm \frac{\sqrt{37}}{\sqrt{50}}$$

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

$$r =\frac{3}{5} - \frac{\sqrt{37}}{\sqrt{50}} = -0.260 s = \frac{3}{5} + \frac{\sqrt{37}}{\sqrt{50}} = 1.460$$

Evaluate

$$150x^{2}-180x-57$$