Solve 7x^2-7x-7

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

Factor

$$7\left(x-\frac{1-\sqrt{5}}{2}\right)\left(x-\frac{\sqrt{5}+1}{2}\right)$$

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.

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

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.

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

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 7\left(-7\right)}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-\left(-7\right)±\sqrt{49-28\left(-7\right)}}{2\times 7}$$

Multiply -28 times -7.

$$x=\frac{-\left(-7\right)±\sqrt{49+196}}{2\times 7}$$

Add 49 to 196.

$$x=\frac{-\left(-7\right)±\sqrt{245}}{2\times 7}$$

Take the square root of 245.

$$x=\frac{-\left(-7\right)±7\sqrt{5}}{2\times 7}$$

The opposite of -7 is 7.

$$x=\frac{7±7\sqrt{5}}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{7±7\sqrt{5}}{14}$$

Now solve the equation x=\frac{7±7\sqrt{5}}{14} when ± is plus. Add 7 to 7\sqrt{5}.

$$x=\frac{7\sqrt{5}+7}{14}$$

Divide 7+7\sqrt{5} by 14.

$$x=\frac{\sqrt{5}+1}{2}$$

Now solve the equation x=\frac{7±7\sqrt{5}}{14} when ± is minus. Subtract 7\sqrt{5} from 7.

$$x=\frac{7-7\sqrt{5}}{14}$$

Divide 7-7\sqrt{5} by 14.

$$x=\frac{1-\sqrt{5}}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1+\sqrt{5}}{2} for x_{1} and \frac{1-\sqrt{5}}{2} for x_{2}.

$$7x^{2}-7x-7=7\left(x-\frac{\sqrt{5}+1}{2}\right)\left(x-\frac{1-\sqrt{5}}{2}\right)$$

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.This is achieved by dividing both sides of the equation by 7

$$x ^ 2 -1x -1 = 0$$

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

$$r + s = 1 rs = -1$$

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

$$r = \frac{1}{2} - u s = \frac{1}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -1

$$(\frac{1}{2} - u) (\frac{1}{2} + u) = -1$$

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

$$\frac{1}{4} - u^2 = -1$$

Simplify the expression by subtracting \frac{1}{4} on both sides

$$-u^2 = -1-\frac{1}{4} = -\frac{5}{4}$$

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

$$u^2 = \frac{5}{4} u = \pm\sqrt{\frac{5}{4}} = \pm \frac{\sqrt{5}}{2}$$

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

$$r =\frac{1}{2} - \frac{\sqrt{5}}{2} = -0.618 s = \frac{1}{2} + \frac{\sqrt{5}}{2} = 1.618$$

Evaluate

$$7\left(x^{2}-x-1\right)$$