Solve x^2-12x-21

$$x ^ { 2 } - 12 x - 21$$

Factor

$$\left(x-\left(6-\sqrt{57}\right)\right)\left(x-\left(\sqrt{57}+6\right)\right)$$

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.

$$x^{2}-12x-21=0$$

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.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\left(-21\right)}}{2}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\left(-21\right)}}{2}$$

Multiply -4 times -21.

$$x=\frac{-\left(-12\right)±\sqrt{144+84}}{2}$$

Add 144 to 84.

$$x=\frac{-\left(-12\right)±\sqrt{228}}{2}$$

Take the square root of 228.

$$x=\frac{-\left(-12\right)±2\sqrt{57}}{2}$$

The opposite of -12 is 12.

$$x=\frac{12±2\sqrt{57}}{2}$$

Now solve the equation x=\frac{12±2\sqrt{57}}{2} when ± is plus. Add 12 to 2\sqrt{57}.

$$x=\frac{2\sqrt{57}+12}{2}$$

Divide 12+2\sqrt{57} by 2.

$$x=\sqrt{57}+6$$

Now solve the equation x=\frac{12±2\sqrt{57}}{2} when ± is minus. Subtract 2\sqrt{57} from 12.

$$x=\frac{12-2\sqrt{57}}{2}$$

Divide 12-2\sqrt{57} by 2.

$$x=6-\sqrt{57}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 6+\sqrt{57} for x_{1} and 6-\sqrt{57} for x_{2}.

$$x^{2}-12x-21=\left(x-\left(\sqrt{57}+6\right)\right)\left(x-\left(6-\sqrt{57}\right)\right)$$

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.

$$x ^ 2 -12x -21 = 0$$

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

$$r + s = 12 rs = -21$$

Two numbers r and s sum up to 12 exactly when the average of the two numbers is \frac{1}{2}*12 = 6. 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>

$$r = 6 - u s = 6 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -21

$$(6 - u) (6 + u) = -21$$

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

$$36 - u^2 = -21$$

Simplify the expression by subtracting 36 on both sides

$$-u^2 = -21-36 = -57$$

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

$$u^2 = 57 u = \pm\sqrt{57} = \pm \sqrt{57}$$

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

$$r =6 - \sqrt{57} = -1.550 s = 6 + \sqrt{57} = 13.550$$

Evaluate

$$x^{2}-12x-21$$