Solve x^2+33x+10.

$$x ^ { 2 } + 33 x + 10 .$$

Factor

$$\left(x-\frac{-\sqrt{1049}-33}{2}\right)\left(x-\frac{\sqrt{1049}-33}{2}\right)$$

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.

$$x^{2}+33x+10=0$$

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.

$$x=\frac{-33±\sqrt{33^{2}-4\times 10}}{2}$$

Square 33.

$$x=\frac{-33±\sqrt{1089-4\times 10}}{2}$$

Multiply -4 times 10.

$$x=\frac{-33±\sqrt{1089-40}}{2}$$

Add 1089 to -40.

$$x=\frac{-33±\sqrt{1049}}{2}$$

Now solve the equation x=\frac{-33±\sqrt{1049}}{2} when ± is plus. Add -33 to \sqrt{1049}.

$$x=\frac{\sqrt{1049}-33}{2}$$

Now solve the equation x=\frac{-33±\sqrt{1049}}{2} when ± is minus. Subtract \sqrt{1049} from -33.

$$x=\frac{-\sqrt{1049}-33}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-33+\sqrt{1049}}{2} for x_{1} and \frac{-33-\sqrt{1049}}{2} for x_{2}.

$$x^{2}+33x+10=\left(x-\frac{\sqrt{1049}-33}{2}\right)\left(x-\frac{-\sqrt{1049}-33}{2}\right)$$

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.

$$x ^ 2 +33x +10 = 0$$

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

$$r + s = -33 rs = 10$$

Two numbers r and s sum up to -33 exactly when the average of the two numbers is \frac{1}{2}*-33 = -\frac{33}{2}. 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>

$$r = -\frac{33}{2} - u s = -\frac{33}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 10

$$(-\frac{33}{2} - u) (-\frac{33}{2} + u) = 10$$

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

$$\frac{1089}{4} - u^2 = 10$$

Simplify the expression by subtracting \frac{1089}{4} on both sides

$$-u^2 = 10-\frac{1089}{4} = -\frac{1049}{4}$$

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

$$u^2 = \frac{1049}{4} u = \pm\sqrt{\frac{1049}{4}} = \pm \frac{\sqrt{1049}}{2}$$

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

$$r =-\frac{33}{2} - \frac{\sqrt{1049}}{2} = -32.694 s = -\frac{33}{2} + \frac{\sqrt{1049}}{2} = -0.306$$

Evaluate

$$x^{2}+33x+10$$