Solve 3-4t+t^2

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

Factor

$$\left(t-3\right)\left(t-1\right)$$

Solution Steps

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

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

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

$$a+b=-4 ab=1\times 3=3$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. The only such pair is the system solution.

$$a=-3 b=-1$$

Rewrite t^{2}-4t+3 as \left(t^{2}-3t\right)+\left(-t+3\right).

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

Factor out t in the first and -1 in the second group.

$$t\left(t-3\right)-\left(t-3\right)$$

Factor out common term t-3 by using distributive property.

$$\left(t-3\right)\left(t-1\right)$$

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.

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

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.

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

Square -4.

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

Multiply -4 times 3.

$$t=\frac{-\left(-4\right)±\sqrt{16-12}}{2}$$

Add 16 to -12.

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

Take the square root of 4.

$$t=\frac{-\left(-4\right)±2}{2}$$

The opposite of -4 is 4.

$$t=\frac{4±2}{2}$$

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

$$t=\frac{6}{2}$$

Divide 6 by 2.

$$t=3$$

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

$$t=\frac{2}{2}$$

Divide 2 by 2.

$$t=1$$

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

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

Evaluate

$$\left(t-3\right)\left(t-1\right)$$