Solve 4m^2+8m-20

$$4 m ^ { 2 } + 8 m - 20$$

Factor

$$4\left(m-\left(-\sqrt{6}-1\right)\right)\left(m-\left(\sqrt{6}-1\right)\right)$$

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.

$$4m^{2}+8m-20=0$$

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.

$$m=\frac{-8±\sqrt{8^{2}-4\times 4\left(-20\right)}}{2\times 4}$$

Square 8.

$$m=\frac{-8±\sqrt{64-4\times 4\left(-20\right)}}{2\times 4}$$

Multiply -4 times 4.

$$m=\frac{-8±\sqrt{64-16\left(-20\right)}}{2\times 4}$$

Multiply -16 times -20.

$$m=\frac{-8±\sqrt{64+320}}{2\times 4}$$

Add 64 to 320.

$$m=\frac{-8±\sqrt{384}}{2\times 4}$$

Take the square root of 384.

$$m=\frac{-8±8\sqrt{6}}{2\times 4}$$

Multiply 2 times 4.

$$m=\frac{-8±8\sqrt{6}}{8}$$

Now solve the equation m=\frac{-8±8\sqrt{6}}{8} when ± is plus. Add -8 to 8\sqrt{6}.

$$m=\frac{8\sqrt{6}-8}{8}$$

Divide -8+8\sqrt{6} by 8.

$$m=\sqrt{6}-1$$

Now solve the equation m=\frac{-8±8\sqrt{6}}{8} when ± is minus. Subtract 8\sqrt{6} from -8.

$$m=\frac{-8\sqrt{6}-8}{8}$$

Divide -8-8\sqrt{6} by 8.

$$m=-\sqrt{6}-1$$

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

$$4m^{2}+8m-20=4\left(m-\left(\sqrt{6}-1\right)\right)\left(m-\left(-\sqrt{6}-1\right)\right)$$

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.This is achieved by dividing both sides of the equation by 4

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

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

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

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 = -5

$$(-1 - u) (-1 + u) = -5$$

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

$$1 - u^2 = -5$$

Simplify the expression by subtracting 1 on both sides

$$-u^2 = -5-1 = -6$$

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

$$u^2 = 6 u = \pm\sqrt{6} = \pm \sqrt{6}$$

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

$$r =-1 - \sqrt{6} = -3.449 s = -1 + \sqrt{6} = 1.449$$

Evaluate

$$4\left(m^{2}+2m-5\right)$$