Solve -6+4x+2x^2

$$- 6 + 4 x + 2 x ^ { 2 }$$

Factor

$$2\left(x-1\right)\left(x+3\right)$$

Solution Steps

Factor out 2.

$$2\left(-3+2x+x^{2}\right)$$

Consider -3+2x+x^{2}. Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

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

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-3. To find a and b, set up a system to be solved.

$$a+b=2 ab=1\left(-3\right)=-3$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. The only such pair is the system solution.

$$a=-1 b=3$$

Rewrite x^{2}+2x-3 as \left(x^{2}-x\right)+\left(3x-3\right).

$$\left(x^{2}-x\right)+\left(3x-3\right)$$

Factor out x in the first and 3 in the second group.

$$x\left(x-1\right)+3\left(x-1\right)$$

Factor out common term x-1 by using distributive property.

$$\left(x-1\right)\left(x+3\right)$$

Rewrite the complete factored expression.

$$2\left(x-1\right)\left(x+3\right)$$

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.

$$2x^{2}+4x-6=0$$

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.

$$x=\frac{-4±\sqrt{4^{2}-4\times 2\left(-6\right)}}{2\times 2}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\times 2\left(-6\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-4±\sqrt{16-8\left(-6\right)}}{2\times 2}$$

Multiply -8 times -6.

$$x=\frac{-4±\sqrt{16+48}}{2\times 2}$$

Add 16 to 48.

$$x=\frac{-4±\sqrt{64}}{2\times 2}$$

Take the square root of 64.

$$x=\frac{-4±8}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-4±8}{4}$$

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

$$x=\frac{4}{4}$$

Divide 4 by 4.

$$x=1$$

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

$$x=-\frac{12}{4}$$

Divide -12 by 4.

$$x=-3$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 1 for x_{1} and -3 for x_{2}.

$$2x^{2}+4x-6=2\left(x-1\right)\left(x-\left(-3\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$2x^{2}+4x-6=2\left(x-1\right)\left(x+3\right)$$

Evaluate

$$2\left(x-1\right)\left(x+3\right)$$