Solve 7x^2+12x-420

$$7 x ^ { 2 } + 12 x - 420$$

Factor

$$7\left(x-\frac{-4\sqrt{186}-6}{7}\right)\left(x-\frac{4\sqrt{186}-6}{7}\right)$$

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.

$$7x^{2}+12x-420=0$$

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.

$$x=\frac{-12±\sqrt{12^{2}-4\times 7\left(-420\right)}}{2\times 7}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\times 7\left(-420\right)}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-12±\sqrt{144-28\left(-420\right)}}{2\times 7}$$

Multiply -28 times -420.

$$x=\frac{-12±\sqrt{144+11760}}{2\times 7}$$

Add 144 to 11760.

$$x=\frac{-12±\sqrt{11904}}{2\times 7}$$

Take the square root of 11904.

$$x=\frac{-12±8\sqrt{186}}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{-12±8\sqrt{186}}{14}$$

Now solve the equation x=\frac{-12±8\sqrt{186}}{14} when ± is plus. Add -12 to 8\sqrt{186}.

$$x=\frac{8\sqrt{186}-12}{14}$$

Divide -12+8\sqrt{186} by 14.

$$x=\frac{4\sqrt{186}-6}{7}$$

Now solve the equation x=\frac{-12±8\sqrt{186}}{14} when ± is minus. Subtract 8\sqrt{186} from -12.

$$x=\frac{-8\sqrt{186}-12}{14}$$

Divide -12-8\sqrt{186} by 14.

$$x=\frac{-4\sqrt{186}-6}{7}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-6+4\sqrt{186}}{7} for x_{1} and \frac{-6-4\sqrt{186}}{7} for x_{2}.

$$7x^{2}+12x-420=7\left(x-\frac{4\sqrt{186}-6}{7}\right)\left(x-\frac{-4\sqrt{186}-6}{7}\right)$$

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

$$x ^ 2 +\frac{12}{7}x -60 = 0$$

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

$$r + s = -\frac{12}{7} rs = -60$$

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 = -60

$$(-\frac{6}{7} - u) (-\frac{6}{7} + u) = -60$$

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

$$\frac{36}{49} - u^2 = -60$$

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

$$-u^2 = -60-\frac{36}{49} = -\frac{2976}{49}$$

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

$$u^2 = \frac{2976}{49} u = \pm\sqrt{\frac{2976}{49}} = \pm \frac{\sqrt{2976}}{7}$$

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

$$r =-\frac{6}{7} - \frac{\sqrt{2976}}{7} = -8.650 s = -\frac{6}{7} + \frac{\sqrt{2976}}{7} = 6.936$$

Evaluate

$$7x^{2}+12x-420$$