Solve x^2+42x+15

$$x ^ { 2 } + 42 x + 15$$

Factor

$$\left(x-\left(-\sqrt{426}-21\right)\right)\left(x-\left(\sqrt{426}-21\right)\right)$$

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.

$$x^{2}+42x+15=0$$

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.

$$x=\frac{-42±\sqrt{42^{2}-4\times 15}}{2}$$

Square 42.

$$x=\frac{-42±\sqrt{1764-4\times 15}}{2}$$

Multiply -4 times 15.

$$x=\frac{-42±\sqrt{1764-60}}{2}$$

Add 1764 to -60.

$$x=\frac{-42±\sqrt{1704}}{2}$$

Take the square root of 1704.

$$x=\frac{-42±2\sqrt{426}}{2}$$

Now solve the equation x=\frac{-42±2\sqrt{426}}{2} when ± is plus. Add -42 to 2\sqrt{426}.

$$x=\frac{2\sqrt{426}-42}{2}$$

Divide -42+2\sqrt{426} by 2.

$$x=\sqrt{426}-21$$

Now solve the equation x=\frac{-42±2\sqrt{426}}{2} when ± is minus. Subtract 2\sqrt{426} from -42.

$$x=\frac{-2\sqrt{426}-42}{2}$$

Divide -42-2\sqrt{426} by 2.

$$x=-\sqrt{426}-21$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -21+\sqrt{426} for x_{1} and -21-\sqrt{426} for x_{2}.

$$x^{2}+42x+15=\left(x-\left(\sqrt{426}-21\right)\right)\left(x-\left(-\sqrt{426}-21\right)\right)$$

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.

$$x ^ 2 +42x +15 = 0$$

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

$$r + s = -42 rs = 15$$

Two numbers r and s sum up to -42 exactly when the average of the two numbers is \frac{1}{2}*-42 = -21. 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>

$$r = -21 - u s = -21 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 15

$$(-21 - u) (-21 + u) = 15$$

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

$$441 - u^2 = 15$$

Simplify the expression by subtracting 441 on both sides

$$-u^2 = 15-441 = -426$$

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

$$u^2 = 426 u = \pm\sqrt{426} = \pm \sqrt{426}$$

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

$$r =-21 - \sqrt{426} = -41.640 s = -21 + \sqrt{426} = -0.360$$

Evaluate

$$x^{2}+42x+15$$