Solve 25x^2+50x-6000=

$$25 x ^ { 2 } + 50 x - 6000 =$$

Factor

$$25\left(x-\left(-\sqrt{241}-1\right)\right)\left(x-\left(\sqrt{241}-1\right)\right)$$

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.

$$25x^{2}+50x-6000=0$$

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.

$$x=\frac{-50±\sqrt{50^{2}-4\times 25\left(-6000\right)}}{2\times 25}$$

Square 50.

$$x=\frac{-50±\sqrt{2500-4\times 25\left(-6000\right)}}{2\times 25}$$

Multiply -4 times 25.

$$x=\frac{-50±\sqrt{2500-100\left(-6000\right)}}{2\times 25}$$

Multiply -100 times -6000.

$$x=\frac{-50±\sqrt{2500+600000}}{2\times 25}$$

Add 2500 to 600000.

$$x=\frac{-50±\sqrt{602500}}{2\times 25}$$

Take the square root of 602500.

$$x=\frac{-50±50\sqrt{241}}{2\times 25}$$

Multiply 2 times 25.

$$x=\frac{-50±50\sqrt{241}}{50}$$

Now solve the equation x=\frac{-50±50\sqrt{241}}{50} when ± is plus. Add -50 to 50\sqrt{241}.

$$x=\frac{50\sqrt{241}-50}{50}$$

Divide -50+50\sqrt{241} by 50.

$$x=\sqrt{241}-1$$

Now solve the equation x=\frac{-50±50\sqrt{241}}{50} when ± is minus. Subtract 50\sqrt{241} from -50.

$$x=\frac{-50\sqrt{241}-50}{50}$$

Divide -50-50\sqrt{241} by 50.

$$x=-\sqrt{241}-1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -1+\sqrt{241} for x_{1} and -1-\sqrt{241} for x_{2}.

$$25x^{2}+50x-6000=25\left(x-\left(\sqrt{241}-1\right)\right)\left(x-\left(-\sqrt{241}-1\right)\right)$$

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 25

$$x ^ 2 +2x -240 = 0$$

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

$$r + s = -2 rs = -240$$

Two numbers r and s sum up to -2 exactly when the average of the two numbers is \frac{1}{2}*-2 = -1. 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-gzdabgg4ehffg0hf.b01.azurefd.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = -1 - u s = -1 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -240

$$(-1 - u) (-1 + u) = -240$$

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

$$1 - u^2 = -240$$

Simplify the expression by subtracting 1 on both sides

$$-u^2 = -240-1 = -241$$

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

$$u^2 = 241 u = \pm\sqrt{241} = \pm \sqrt{241}$$

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

$$r =-1 - \sqrt{241} = -16.524 s = -1 + \sqrt{241} = 14.524$$

Evaluate

$$25\left(x^{2}+2x-240\right)$$