Solve x^2-30x-450

$$x ^ { 2 } - 30 x - 450$$

Factor

$$\left(x-\left(15-15\sqrt{3}\right)\right)\left(x-\left(15\sqrt{3}+15\right)\right)$$

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.

$$x^{2}-30x-450=0$$

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.

$$x=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\left(-450\right)}}{2}$$

Square -30.

$$x=\frac{-\left(-30\right)±\sqrt{900-4\left(-450\right)}}{2}$$

Multiply -4 times -450.

$$x=\frac{-\left(-30\right)±\sqrt{900+1800}}{2}$$

Add 900 to 1800.

$$x=\frac{-\left(-30\right)±\sqrt{2700}}{2}$$

Take the square root of 2700.

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

The opposite of -30 is 30.

$$x=\frac{30±30\sqrt{3}}{2}$$

Now solve the equation x=\frac{30±30\sqrt{3}}{2} when ± is plus. Add 30 to 30\sqrt{3}.

$$x=\frac{30\sqrt{3}+30}{2}$$

Divide 30+30\sqrt{3} by 2.

$$x=15\sqrt{3}+15$$

Now solve the equation x=\frac{30±30\sqrt{3}}{2} when ± is minus. Subtract 30\sqrt{3} from 30.

$$x=\frac{30-30\sqrt{3}}{2}$$

Divide 30-30\sqrt{3} by 2.

$$x=15-15\sqrt{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 15+15\sqrt{3} for x_{1} and 15-15\sqrt{3} for x_{2}.

$$x^{2}-30x-450=\left(x-\left(15\sqrt{3}+15\right)\right)\left(x-\left(15-15\sqrt{3}\right)\right)$$

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.

$$x ^ 2 -30x -450 = 0$$

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

$$r + s = 30 rs = -450$$

Two numbers r and s sum up to 30 exactly when the average of the two numbers is \frac{1}{2}*30 = 15. 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>

$$r = 15 - u s = 15 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -450

$$(15 - u) (15 + u) = -450$$

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

$$225 - u^2 = -450$$

Simplify the expression by subtracting 225 on both sides

$$-u^2 = -450-225 = -675$$

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

$$u^2 = 675 u = \pm\sqrt{675} = \pm \sqrt{675}$$

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

$$r =15 - \sqrt{675} = -10.981 s = 15 + \sqrt{675} = 40.981$$

Evaluate

$$x^{2}-30x-450$$