Solve -3x^2-300x+75

$$- 3 x ^ { 2 } - 300 x + 75$$

Factor

$$-3\left(x-\left(-5\sqrt{101}-50\right)\right)\left(x-\left(5\sqrt{101}-50\right)\right)$$

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.

$$-3x^{2}-300x+75=0$$

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.

$$x=\frac{-\left(-300\right)±\sqrt{\left(-300\right)^{2}-4\left(-3\right)\times 75}}{2\left(-3\right)}$$

Square -300.

$$x=\frac{-\left(-300\right)±\sqrt{90000-4\left(-3\right)\times 75}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-\left(-300\right)±\sqrt{90000+12\times 75}}{2\left(-3\right)}$$

Multiply 12 times 75.

$$x=\frac{-\left(-300\right)±\sqrt{90000+900}}{2\left(-3\right)}$$

Add 90000 to 900.

$$x=\frac{-\left(-300\right)±\sqrt{90900}}{2\left(-3\right)}$$

Take the square root of 90900.

$$x=\frac{-\left(-300\right)±30\sqrt{101}}{2\left(-3\right)}$$

The opposite of -300 is 300.

$$x=\frac{300±30\sqrt{101}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{300±30\sqrt{101}}{-6}$$

Now solve the equation x=\frac{300±30\sqrt{101}}{-6} when ± is plus. Add 300 to 30\sqrt{101}.

$$x=\frac{30\sqrt{101}+300}{-6}$$

Divide 300+30\sqrt{101} by -6.

$$x=-5\sqrt{101}-50$$

Now solve the equation x=\frac{300±30\sqrt{101}}{-6} when ± is minus. Subtract 30\sqrt{101} from 300.

$$x=\frac{300-30\sqrt{101}}{-6}$$

Divide 300-30\sqrt{101} by -6.

$$x=5\sqrt{101}-50$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -50-5\sqrt{101} for x_{1} and -50+5\sqrt{101} for x_{2}.

$$-3x^{2}-300x+75=-3\left(x-\left(-5\sqrt{101}-50\right)\right)\left(x-\left(5\sqrt{101}-50\right)\right)$$

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.

$$x ^ 2 +100x -25 = 0$$

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

$$r + s = -100 rs = -25$$

Two numbers r and s sum up to -100 exactly when the average of the two numbers is \frac{1}{2}*-100 = -50. 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>

$$r = -50 - u s = -50 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -25

$$(-50 - u) (-50 + u) = -25$$

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

$$2500 - u^2 = -25$$

Simplify the expression by subtracting 2500 on both sides

$$-u^2 = -25-2500 = -2525$$

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

$$u^2 = 2525 u = \pm\sqrt{2525} = \pm \sqrt{2525}$$

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

$$r =-50 - \sqrt{2525} = -100.249 s = -50 + \sqrt{2525} = 0.249$$

Evaluate

$$75-300x-3x^{2}$$