Solve -3x^2+2x+1

$$-3 { x }^{ 2 } +2x+1$$

Factor

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

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -3x^{2}+ax+bx+1. To find a and b, set up a system to be solved.

$$a+b=2 ab=-3=-3$$

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.

$$a=3 b=-1$$

Rewrite -3x^{2}+2x+1 as \left(-3x^{2}+3x\right)+\left(-x+1\right).

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

Factor out 3x in -3x^{2}+3x.

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

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

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

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.

$$-3x^{2}+2x+1=0$$

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.

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

Square 2.

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

Multiply -4 times -3.

$$x=\frac{-2±\sqrt{4+12}}{2\left(-3\right)}$$

Add 4 to 12.

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

Take the square root of 16.

$$x=\frac{-2±4}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-2±4}{-6}$$

Now solve the equation x=\frac{-2±4}{-6} when ± is plus. Add -2 to 4.

$$x=\frac{2}{-6}$$

Reduce the fraction \frac{2}{-6} to lowest terms by extracting and canceling out 2.

$$x=-\frac{1}{3}$$

Now solve the equation x=\frac{-2±4}{-6} when ± is minus. Subtract 4 from -2.

$$x=-\frac{6}{-6}$$

Divide -6 by -6.

$$x=1$$

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

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

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

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

Add \frac{1}{3} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

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

Cancel out 3, the greatest common factor in -3 and 3.

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

Evaluate

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