Solve 7x^2+35x-14

$$7 x ^ { 2 } + 35 x - 14$$

Factor

$$7\left(x-\frac{-\sqrt{33}-5}{2}\right)\left(x-\frac{\sqrt{33}-5}{2}\right)$$

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.

$$7x^{2}+35x-14=0$$

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.

$$x=\frac{-35±\sqrt{35^{2}-4\times 7\left(-14\right)}}{2\times 7}$$

Square 35.

$$x=\frac{-35±\sqrt{1225-4\times 7\left(-14\right)}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-35±\sqrt{1225-28\left(-14\right)}}{2\times 7}$$

Multiply -28 times -14.

$$x=\frac{-35±\sqrt{1225+392}}{2\times 7}$$

Add 1225 to 392.

$$x=\frac{-35±\sqrt{1617}}{2\times 7}$$

Take the square root of 1617.

$$x=\frac{-35±7\sqrt{33}}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{-35±7\sqrt{33}}{14}$$

Now solve the equation x=\frac{-35±7\sqrt{33}}{14} when ± is plus. Add -35 to 7\sqrt{33}.

$$x=\frac{7\sqrt{33}-35}{14}$$

Divide -35+7\sqrt{33} by 14.

$$x=\frac{\sqrt{33}-5}{2}$$

Now solve the equation x=\frac{-35±7\sqrt{33}}{14} when ± is minus. Subtract 7\sqrt{33} from -35.

$$x=\frac{-7\sqrt{33}-35}{14}$$

Divide -35-7\sqrt{33} by 14.

$$x=\frac{-\sqrt{33}-5}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-5+\sqrt{33}}{2} for x_{1} and \frac{-5-\sqrt{33}}{2} for x_{2}.

$$7x^{2}+35x-14=7\left(x-\frac{\sqrt{33}-5}{2}\right)\left(x-\frac{-\sqrt{33}-5}{2}\right)$$

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

$$x ^ 2 +5x -2 = 0$$

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

$$r + s = -5 rs = -2$$

Two numbers r and s sum up to -5 exactly when the average of the two numbers is \frac{1}{2}*-5 = -\frac{5}{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>

$$r = -\frac{5}{2} - u s = -\frac{5}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -2

$$(-\frac{5}{2} - u) (-\frac{5}{2} + u) = -2$$

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

$$\frac{25}{4} - u^2 = -2$$

Simplify the expression by subtracting \frac{25}{4} on both sides

$$-u^2 = -2-\frac{25}{4} = -\frac{33}{4}$$

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

$$u^2 = \frac{33}{4} u = \pm\sqrt{\frac{33}{4}} = \pm \frac{\sqrt{33}}{2}$$

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

$$r =-\frac{5}{2} - \frac{\sqrt{33}}{2} = -5.372 s = -\frac{5}{2} + \frac{\sqrt{33}}{2} = 0.372$$

Evaluate

$$7\left(x^{2}+5x-2\right)$$