Solve 7x^2-12x-85=

$$7 x ^ { 2 } - 12 x - 85 =$$

Factor

$$7\left(x-\frac{6-\sqrt{631}}{7}\right)\left(x-\frac{\sqrt{631}+6}{7}\right)$$

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.

$$7x^{2}-12x-85=0$$

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.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 7\left(-85\right)}}{2\times 7}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 7\left(-85\right)}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-\left(-12\right)±\sqrt{144-28\left(-85\right)}}{2\times 7}$$

Multiply -28 times -85.

$$x=\frac{-\left(-12\right)±\sqrt{144+2380}}{2\times 7}$$

Add 144 to 2380.

$$x=\frac{-\left(-12\right)±\sqrt{2524}}{2\times 7}$$

Take the square root of 2524.

$$x=\frac{-\left(-12\right)±2\sqrt{631}}{2\times 7}$$

The opposite of -12 is 12.

$$x=\frac{12±2\sqrt{631}}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{12±2\sqrt{631}}{14}$$

Now solve the equation x=\frac{12±2\sqrt{631}}{14} when ± is plus. Add 12 to 2\sqrt{631}.

$$x=\frac{2\sqrt{631}+12}{14}$$

Divide 12+2\sqrt{631} by 14.

$$x=\frac{\sqrt{631}+6}{7}$$

Now solve the equation x=\frac{12±2\sqrt{631}}{14} when ± is minus. Subtract 2\sqrt{631} from 12.

$$x=\frac{12-2\sqrt{631}}{14}$$

Divide 12-2\sqrt{631} by 14.

$$x=\frac{6-\sqrt{631}}{7}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{6+\sqrt{631}}{7} for x_{1} and \frac{6-\sqrt{631}}{7} for x_{2}.

$$7x^{2}-12x-85=7\left(x-\frac{\sqrt{631}+6}{7}\right)\left(x-\frac{6-\sqrt{631}}{7}\right)$$

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 7

$$x ^ 2 -\frac{12}{7}x -\frac{85}{7} = 0$$

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

$$r + s = \frac{12}{7} rs = -\frac{85}{7}$$

Two numbers r and s sum up to \frac{12}{7} exactly when the average of the two numbers is \frac{1}{2}*\frac{12}{7} = \frac{6}{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>

$$r = \frac{6}{7} - u s = \frac{6}{7} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{85}{7}

$$(\frac{6}{7} - u) (\frac{6}{7} + u) = -\frac{85}{7}$$

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

$$\frac{36}{49} - u^2 = -\frac{85}{7}$$

Simplify the expression by subtracting \frac{36}{49} on both sides

$$-u^2 = -\frac{85}{7}-\frac{36}{49} = -\frac{631}{49}$$

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

$$u^2 = \frac{631}{49} u = \pm\sqrt{\frac{631}{49}} = \pm \frac{\sqrt{631}}{7}$$

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

$$r =\frac{6}{7} - \frac{\sqrt{631}}{7} = -2.731 s = \frac{6}{7} + \frac{\sqrt{631}}{7} = 4.446$$

Evaluate

$$7x^{2}-12x-85$$