Solve -2x^2+4x+3

$$- 2 x ^ { 2 } + 4 x + 3$$

Factor

$$-2\left(x-\left(-\frac{\sqrt{10}}{2}+1\right)\right)\left(x-\left(\frac{\sqrt{10}}{2}+1\right)\right)$$

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.

$$-2x^{2}+4x+3=0$$

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.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-2\right)\times 3}}{2\left(-2\right)}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-2\right)\times 3}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-4±\sqrt{16+8\times 3}}{2\left(-2\right)}$$

Multiply 8 times 3.

$$x=\frac{-4±\sqrt{16+24}}{2\left(-2\right)}$$

Add 16 to 24.

$$x=\frac{-4±\sqrt{40}}{2\left(-2\right)}$$

Take the square root of 40.

$$x=\frac{-4±2\sqrt{10}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-4±2\sqrt{10}}{-4}$$

Now solve the equation x=\frac{-4±2\sqrt{10}}{-4} when ± is plus. Add -4 to 2\sqrt{10}.

$$x=\frac{2\sqrt{10}-4}{-4}$$

Divide -4+2\sqrt{10} by -4.

$$x=-\frac{\sqrt{10}}{2}+1$$

Now solve the equation x=\frac{-4±2\sqrt{10}}{-4} when ± is minus. Subtract 2\sqrt{10} from -4.

$$x=\frac{-2\sqrt{10}-4}{-4}$$

Divide -4-2\sqrt{10} by -4.

$$x=\frac{\sqrt{10}}{2}+1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 1-\frac{\sqrt{10}}{2} for x_{1} and 1+\frac{\sqrt{10}}{2} for x_{2}.

$$-2x^{2}+4x+3=-2\left(x-\left(-\frac{\sqrt{10}}{2}+1\right)\right)\left(x-\left(\frac{\sqrt{10}}{2}+1\right)\right)$$

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.

$$x ^ 2 -2x -\frac{3}{2} = 0$$

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

$$r + s = 2 rs = -\frac{3}{2}$$

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 = -\frac{3}{2}

$$(1 - u) (1 + u) = -\frac{3}{2}$$

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

$$1 - u^2 = -\frac{3}{2}$$

Simplify the expression by subtracting 1 on both sides

$$-u^2 = -\frac{3}{2}-1 = -\frac{5}{2}$$

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

$$u^2 = \frac{5}{2} u = \pm\sqrt{\frac{5}{2}} = \pm \frac{\sqrt{5}}{\sqrt{2}}$$

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

$$r =1 - \frac{\sqrt{5}}{\sqrt{2}} = -0.581 s = 1 + \frac{\sqrt{5}}{\sqrt{2}} = 2.581$$

Evaluate

$$3+4x-2x^{2}$$