Solve f(3)=2x^2-5x(3)+2

$$f ( 3 ) = 2 x ^ { 2 } - 5 x ( 3 ) + 2$$

Factor

$$2\left(x-\frac{15-\sqrt{209}}{4}\right)\left(x-\frac{\sqrt{209}+15}{4}\right)$$

Steps Using the Quadratic Formula

Multiply 5 and 3 to get 15.

$$factor(2x^{2}-15x+2)$$

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.

$$2x^{2}-15x+2=0$$

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.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times 2\times 2}}{2\times 2}$$

Square -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\times 2\times 2}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-15\right)±\sqrt{225-8\times 2}}{2\times 2}$$

Multiply -8 times 2.

$$x=\frac{-\left(-15\right)±\sqrt{225-16}}{2\times 2}$$

Add 225 to -16.

$$x=\frac{-\left(-15\right)±\sqrt{209}}{2\times 2}$$

The opposite of -15 is 15.

$$x=\frac{15±\sqrt{209}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{15±\sqrt{209}}{4}$$

Now solve the equation x=\frac{15±\sqrt{209}}{4} when ± is plus. Add 15 to \sqrt{209}.

$$x=\frac{\sqrt{209}+15}{4}$$

Now solve the equation x=\frac{15±\sqrt{209}}{4} when ± is minus. Subtract \sqrt{209} from 15.

$$x=\frac{15-\sqrt{209}}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{15+\sqrt{209}}{4} for x_{1} and \frac{15-\sqrt{209}}{4} for x_{2}.

$$2x^{2}-15x+2=2\left(x-\frac{\sqrt{209}+15}{4}\right)\left(x-\frac{15-\sqrt{209}}{4}\right)$$

Evaluate

$$2x^{2}-15x+2$$

Solution Steps

Multiply 5 and 3 to get 15.

$$2x^{2}-15x+2$$