Solve 49x^2+126x+9

$$49 x ^ { 2 } + 126 x + 9$$

Factor

$$49\left(x-\frac{-6\sqrt{2}-9}{7}\right)\left(x-\frac{6\sqrt{2}-9}{7}\right)$$

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.

$$49x^{2}+126x+9=0$$

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.

$$x=\frac{-126±\sqrt{126^{2}-4\times 49\times 9}}{2\times 49}$$

Square 126.

$$x=\frac{-126±\sqrt{15876-4\times 49\times 9}}{2\times 49}$$

Multiply -4 times 49.

$$x=\frac{-126±\sqrt{15876-196\times 9}}{2\times 49}$$

Multiply -196 times 9.

$$x=\frac{-126±\sqrt{15876-1764}}{2\times 49}$$

Add 15876 to -1764.

$$x=\frac{-126±\sqrt{14112}}{2\times 49}$$

Take the square root of 14112.

$$x=\frac{-126±84\sqrt{2}}{2\times 49}$$

Multiply 2 times 49.

$$x=\frac{-126±84\sqrt{2}}{98}$$

Now solve the equation x=\frac{-126±84\sqrt{2}}{98} when ± is plus. Add -126 to 84\sqrt{2}.

$$x=\frac{84\sqrt{2}-126}{98}$$

Divide -126+84\sqrt{2} by 98.

$$x=\frac{6\sqrt{2}-9}{7}$$

Now solve the equation x=\frac{-126±84\sqrt{2}}{98} when ± is minus. Subtract 84\sqrt{2} from -126.

$$x=\frac{-84\sqrt{2}-126}{98}$$

Divide -126-84\sqrt{2} by 98.

$$x=\frac{-6\sqrt{2}-9}{7}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-9+6\sqrt{2}}{7} for x_{1} and \frac{-9-6\sqrt{2}}{7} for x_{2}.

$$49x^{2}+126x+9=49\left(x-\frac{6\sqrt{2}-9}{7}\right)\left(x-\frac{-6\sqrt{2}-9}{7}\right)$$

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.This is achieved by dividing both sides of the equation by 49

$$x ^ 2 +\frac{18}{7}x +\frac{9}{49} = 0$$

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

$$r + s = -\frac{18}{7} rs = \frac{9}{49}$$

Two numbers r and s sum up to -\frac{18}{7} exactly when the average of the two numbers is \frac{1}{2}*-\frac{18}{7} = -\frac{9}{7}. 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>

$$r = -\frac{9}{7} - u s = -\frac{9}{7} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{9}{49}

$$(-\frac{9}{7} - u) (-\frac{9}{7} + u) = \frac{9}{49}$$

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

$$\frac{81}{49} - u^2 = \frac{9}{49}$$

Simplify the expression by subtracting \frac{81}{49} on both sides

$$-u^2 = \frac{9}{49}-\frac{81}{49} = -\frac{72}{49}$$

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

$$u^2 = \frac{72}{49} u = \pm\sqrt{\frac{72}{49}} = \pm \frac{\sqrt{72}}{7}$$

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

$$r =-\frac{9}{7} - \frac{\sqrt{72}}{7} = -2.498 s = -\frac{9}{7} + \frac{\sqrt{72}}{7} = -0.074$$

Evaluate

$$49x^{2}+126x+9$$