Solve f(x)=-7x^2-14x+7

$$f ( x ) = - 7 x ^ { 2 } - 14 x + 7$$

Factor

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

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.

$$-7x^{2}-14x+7=0$$

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.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\left(-7\right)\times 7}}{2\left(-7\right)}$$

Square -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\left(-7\right)\times 7}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$x=\frac{-\left(-14\right)±\sqrt{196+28\times 7}}{2\left(-7\right)}$$

Multiply 28 times 7.

$$x=\frac{-\left(-14\right)±\sqrt{196+196}}{2\left(-7\right)}$$

Add 196 to 196.

$$x=\frac{-\left(-14\right)±\sqrt{392}}{2\left(-7\right)}$$

Take the square root of 392.

$$x=\frac{-\left(-14\right)±14\sqrt{2}}{2\left(-7\right)}$$

The opposite of -14 is 14.

$$x=\frac{14±14\sqrt{2}}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{14±14\sqrt{2}}{-14}$$

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

$$x=\frac{14\sqrt{2}+14}{-14}$$

Divide 14+14\sqrt{2} by -14.

$$x=-\left(\sqrt{2}+1\right)$$

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

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

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

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

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

$$-7x^{2}-14x+7=-7\left(x-\left(-\left(\sqrt{2}+1\right)\right)\right)\left(x-\left(\sqrt{2}-1\right)\right)$$

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.

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

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

$$r + s = -2 rs = -1$$

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

$$r = -1 - u s = -1 + u$$

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

$$(-1 - u) (-1 + u) = -1$$

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

$$1 - u^2 = -1$$

Simplify the expression by subtracting 1 on both sides

$$-u^2 = -1-1 = -2$$

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

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

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

$$r =-1 - \sqrt{2} = -2.414 s = -1 + \sqrt{2} = 0.414$$

Evaluate

$$7-14x-7x^{2}$$