Solve 21x^2-9x+3

$$21 { x }^{ 2 } -9x+3$$

Factor

$$3\left(7x^{2}-3x+1\right)$$

Solution Steps

Factor out 3. Polynomial 7x^{2}-3x+1 is not factored since it does not have any rational roots.

$$3\left(7x^{2}-3x+1\right)$$

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.

$$21x^{2}-9x+3=0$$

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.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 21\times 3}}{2\times 21}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 21\times 3}}{2\times 21}$$

Multiply -4 times 21.

$$x=\frac{-\left(-9\right)±\sqrt{81-84\times 3}}{2\times 21}$$

Multiply -84 times 3.

$$x=\frac{-\left(-9\right)±\sqrt{81-252}}{2\times 21}$$

Add 81 to -252.

$$x=\frac{-\left(-9\right)±\sqrt{-171}}{2\times 21}$$

Since the square root of a negative number is not defined in the real field, there are no solutions. Quadratic polynomial cannot be factored.

$$21x^{2}-9x+3$$

Evaluate

$$21x^{2}-9x+3$$