Solve 2x^2-3x-5*2x+3

$$2 { x }^{ 2 } -3x-5 \times 2x+3$$

Evaluate

$$2x^{2}-13x+3$$

Solution Steps

Multiply 5 and 2 to get 10.

$$2x^{2}-3x-10x+3$$

Combine -3x and -10x to get -13x.

$$2x^{2}-13x+3$$

Factor

$$2\left(x-\frac{13-\sqrt{145}}{4}\right)\left(x-\frac{\sqrt{145}+13}{4}\right)$$

Steps Using the Quadratic Formula

Multiply 5 and 2 to get 10.

$$factor(2x^{2}-3x-10x+3)$$

Combine -3x and -10x to get -13x.

$$factor(2x^{2}-13x+3)$$

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.

$$2x^{2}-13x+3=0$$

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.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 2\times 3}}{2\times 2}$$

Square -13.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\times 2\times 3}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-13\right)±\sqrt{169-8\times 3}}{2\times 2}$$

Multiply -8 times 3.

$$x=\frac{-\left(-13\right)±\sqrt{169-24}}{2\times 2}$$

Add 169 to -24.

$$x=\frac{-\left(-13\right)±\sqrt{145}}{2\times 2}$$

The opposite of -13 is 13.

$$x=\frac{13±\sqrt{145}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{13±\sqrt{145}}{4}$$

Now solve the equation x=\frac{13±\sqrt{145}}{4} when ± is plus. Add 13 to \sqrt{145}.

$$x=\frac{\sqrt{145}+13}{4}$$

Now solve the equation x=\frac{13±\sqrt{145}}{4} when ± is minus. Subtract \sqrt{145} from 13.

$$x=\frac{13-\sqrt{145}}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{13+\sqrt{145}}{4} for x_{1} and \frac{13-\sqrt{145}}{4} for x_{2}.

$$2x^{2}-13x+3=2\left(x-\frac{\sqrt{145}+13}{4}\right)\left(x-\frac{13-\sqrt{145}}{4}\right)$$