Solve f(x)=-7x^2-8x+4

$$f ( x ) = - 7 x ^ { 2 } - 8 x + 4$$

Factor

$$-7\left(x-\frac{-2\sqrt{11}-4}{7}\right)\left(x-\frac{2\sqrt{11}-4}{7}\right)$$

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.

$$-7x^{2}-8x+4=0$$

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.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-7\right)\times 4}}{2\left(-7\right)}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-7\right)\times 4}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$x=\frac{-\left(-8\right)±\sqrt{64+28\times 4}}{2\left(-7\right)}$$

Multiply 28 times 4.

$$x=\frac{-\left(-8\right)±\sqrt{64+112}}{2\left(-7\right)}$$

Add 64 to 112.

$$x=\frac{-\left(-8\right)±\sqrt{176}}{2\left(-7\right)}$$

Take the square root of 176.

$$x=\frac{-\left(-8\right)±4\sqrt{11}}{2\left(-7\right)}$$

The opposite of -8 is 8.

$$x=\frac{8±4\sqrt{11}}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{8±4\sqrt{11}}{-14}$$

Now solve the equation x=\frac{8±4\sqrt{11}}{-14} when ± is plus. Add 8 to 4\sqrt{11}.

$$x=\frac{4\sqrt{11}+8}{-14}$$

Divide 8+4\sqrt{11} by -14.

$$x=\frac{-2\sqrt{11}-4}{7}$$

Now solve the equation x=\frac{8±4\sqrt{11}}{-14} when ± is minus. Subtract 4\sqrt{11} from 8.

$$x=\frac{8-4\sqrt{11}}{-14}$$

Divide 8-4\sqrt{11} by -14.

$$x=\frac{2\sqrt{11}-4}{7}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-4-2\sqrt{11}}{7} for x_{1} and \frac{-4+2\sqrt{11}}{7} for x_{2}.

$$-7x^{2}-8x+4=-7\left(x-\frac{-2\sqrt{11}-4}{7}\right)\left(x-\frac{2\sqrt{11}-4}{7}\right)$$

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.

$$x ^ 2 +\frac{8}{7}x -\frac{4}{7} = 0$$

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

$$r + s = -\frac{8}{7} rs = -\frac{4}{7}$$

Two numbers r and s sum up to -\frac{8}{7} exactly when the average of the two numbers is \frac{1}{2}*-\frac{8}{7} = -\frac{4}{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>

$$r = -\frac{4}{7} - u s = -\frac{4}{7} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{4}{7}

$$(-\frac{4}{7} - u) (-\frac{4}{7} + u) = -\frac{4}{7}$$

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

$$\frac{16}{49} - u^2 = -\frac{4}{7}$$

Simplify the expression by subtracting \frac{16}{49} on both sides

$$-u^2 = -\frac{4}{7}-\frac{16}{49} = -\frac{44}{49}$$

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

$$u^2 = \frac{44}{49} u = \pm\sqrt{\frac{44}{49}} = \pm \frac{\sqrt{44}}{7}$$

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

$$r =-\frac{4}{7} - \frac{\sqrt{44}}{7} = -1.519 s = -\frac{4}{7} + \frac{\sqrt{44}}{7} = 0.376$$

Evaluate

$$4-8x-7x^{2}$$