Solve 20x^2-1000x+120

$$20 x ^ { 2 } - 1000 x + 120$$

Factor

$$20\left(x-\left(25-\sqrt{619}\right)\right)\left(x-\left(\sqrt{619}+25\right)\right)$$

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.

$$20x^{2}-1000x+120=0$$

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.

$$x=\frac{-\left(-1000\right)±\sqrt{\left(-1000\right)^{2}-4\times 20\times 120}}{2\times 20}$$

Square -1000.

$$x=\frac{-\left(-1000\right)±\sqrt{1000000-4\times 20\times 120}}{2\times 20}$$

Multiply -4 times 20.

$$x=\frac{-\left(-1000\right)±\sqrt{1000000-80\times 120}}{2\times 20}$$

Multiply -80 times 120.

$$x=\frac{-\left(-1000\right)±\sqrt{1000000-9600}}{2\times 20}$$

Add 1000000 to -9600.

$$x=\frac{-\left(-1000\right)±\sqrt{990400}}{2\times 20}$$

Take the square root of 990400.

$$x=\frac{-\left(-1000\right)±40\sqrt{619}}{2\times 20}$$

The opposite of -1000 is 1000.

$$x=\frac{1000±40\sqrt{619}}{2\times 20}$$

Multiply 2 times 20.

$$x=\frac{1000±40\sqrt{619}}{40}$$

Now solve the equation x=\frac{1000±40\sqrt{619}}{40} when ± is plus. Add 1000 to 40\sqrt{619}.

$$x=\frac{40\sqrt{619}+1000}{40}$$

Divide 1000+40\sqrt{619} by 40.

$$x=\sqrt{619}+25$$

Now solve the equation x=\frac{1000±40\sqrt{619}}{40} when ± is minus. Subtract 40\sqrt{619} from 1000.

$$x=\frac{1000-40\sqrt{619}}{40}$$

Divide 1000-40\sqrt{619} by 40.

$$x=25-\sqrt{619}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 25+\sqrt{619} for x_{1} and 25-\sqrt{619} for x_{2}.

$$20x^{2}-1000x+120=20\left(x-\left(\sqrt{619}+25\right)\right)\left(x-\left(25-\sqrt{619}\right)\right)$$

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 20

$$x ^ 2 -50x +6 = 0$$

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

$$r + s = 50 rs = 6$$

Two numbers r and s sum up to 50 exactly when the average of the two numbers is \frac{1}{2}*50 = 25. 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>

$$r = 25 - u s = 25 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 6

$$(25 - u) (25 + u) = 6$$

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

$$625 - u^2 = 6$$

Simplify the expression by subtracting 625 on both sides

$$-u^2 = 6-625 = -619$$

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

$$u^2 = 619 u = \pm\sqrt{619} = \pm \sqrt{619}$$

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

$$r =25 - \sqrt{619} = 0.120 s = 25 + \sqrt{619} = 49.880$$

Evaluate

$$20\left(x^{2}-50x+6\right)$$