Solve 21t^2-12t+3

$$21 { t }^{ 2 } -12t+3$$

Factor

$$3\left(7t^{2}-4t+1\right)$$

Solution Steps

Factor out 3. Polynomial 7t^{2}-4t+1 is not factored since it does not have any rational roots.

$$3\left(7t^{2}-4t+1\right)$$

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.

$$21t^{2}-12t+3=0$$

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.

$$t=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 21\times 3}}{2\times 21}$$

Square -12.

$$t=\frac{-\left(-12\right)±\sqrt{144-4\times 21\times 3}}{2\times 21}$$

Multiply -4 times 21.

$$t=\frac{-\left(-12\right)±\sqrt{144-84\times 3}}{2\times 21}$$

Multiply -84 times 3.

$$t=\frac{-\left(-12\right)±\sqrt{144-252}}{2\times 21}$$

Add 144 to -252.

$$t=\frac{-\left(-12\right)±\sqrt{-108}}{2\times 21}$$

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

$$21t^{2}-12t+3$$

Evaluate

$$21t^{2}-12t+3$$