Solve 127x^2+88x+15

$$127 x ^ { 2 } + 88 x + 15$$

Factor

$$127\left(x-\frac{-\sqrt{31}-44}{127}\right)\left(x-\frac{\sqrt{31}-44}{127}\right)$$

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.

$$127x^{2}+88x+15=0$$

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.

$$x=\frac{-88±\sqrt{88^{2}-4\times 127\times 15}}{2\times 127}$$

Square 88.

$$x=\frac{-88±\sqrt{7744-4\times 127\times 15}}{2\times 127}$$

Multiply -4 times 127.

$$x=\frac{-88±\sqrt{7744-508\times 15}}{2\times 127}$$

Multiply -508 times 15.

$$x=\frac{-88±\sqrt{7744-7620}}{2\times 127}$$

Add 7744 to -7620.

$$x=\frac{-88±\sqrt{124}}{2\times 127}$$

Take the square root of 124.

$$x=\frac{-88±2\sqrt{31}}{2\times 127}$$

Multiply 2 times 127.

$$x=\frac{-88±2\sqrt{31}}{254}$$

Now solve the equation x=\frac{-88±2\sqrt{31}}{254} when ± is plus. Add -88 to 2\sqrt{31}.

$$x=\frac{2\sqrt{31}-88}{254}$$

Divide -88+2\sqrt{31} by 254.

$$x=\frac{\sqrt{31}-44}{127}$$

Now solve the equation x=\frac{-88±2\sqrt{31}}{254} when ± is minus. Subtract 2\sqrt{31} from -88.

$$x=\frac{-2\sqrt{31}-88}{254}$$

Divide -88-2\sqrt{31} by 254.

$$x=\frac{-\sqrt{31}-44}{127}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-44+\sqrt{31}}{127} for x_{1} and \frac{-44-\sqrt{31}}{127} for x_{2}.

$$127x^{2}+88x+15=127\left(x-\frac{\sqrt{31}-44}{127}\right)\left(x-\frac{-\sqrt{31}-44}{127}\right)$$

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 127

$$x ^ 2 +\frac{88}{127}x +\frac{15}{127} = 0$$

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

$$r + s = -\frac{88}{127} rs = \frac{15}{127}$$

Two numbers r and s sum up to -\frac{88}{127} exactly when the average of the two numbers is \frac{1}{2}*-\frac{88}{127} = -\frac{44}{127}. 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>

$$r = -\frac{44}{127} - u s = -\frac{44}{127} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{15}{127}

$$(-\frac{44}{127} - u) (-\frac{44}{127} + u) = \frac{15}{127}$$

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

$$\frac{1936}{16129} - u^2 = \frac{15}{127}$$

Simplify the expression by subtracting \frac{1936}{16129} on both sides

$$-u^2 = \frac{15}{127}-\frac{1936}{16129} = -\frac{31}{16129}$$

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

$$u^2 = \frac{31}{16129} u = \pm\sqrt{\frac{31}{16129}} = \pm \frac{\sqrt{31}}{127}$$

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

$$r =-\frac{44}{127} - \frac{\sqrt{31}}{127} = -0.390 s = -\frac{44}{127} + \frac{\sqrt{31}}{127} = -0.303$$

Evaluate

$$127x^{2}+88x+15$$