Solve Q(x)=3x^2+2x^2-5x-1

$$Q ( x ) = 3 x ^ { 2 } + 2 x ^ { 2 } - 5 x - 1$$

Factor

$$5\left(x-\left(-\frac{3\sqrt{5}}{10}+\frac{1}{2}\right)\right)\left(x-\left(\frac{3\sqrt{5}}{10}+\frac{1}{2}\right)\right)$$

Steps Using the Quadratic Formula

Combine 3x^{2} and 2x^{2} to get 5x^{2}.

$$factor(5x^{2}-5x-1)$$

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.

$$5x^{2}-5x-1=0$$

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.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 5\left(-1\right)}}{2\times 5}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 5\left(-1\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-5\right)±\sqrt{25-20\left(-1\right)}}{2\times 5}$$

Multiply -20 times -1.

$$x=\frac{-\left(-5\right)±\sqrt{25+20}}{2\times 5}$$

Add 25 to 20.

$$x=\frac{-\left(-5\right)±\sqrt{45}}{2\times 5}$$

Take the square root of 45.

$$x=\frac{-\left(-5\right)±3\sqrt{5}}{2\times 5}$$

The opposite of -5 is 5.

$$x=\frac{5±3\sqrt{5}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{5±3\sqrt{5}}{10}$$

Now solve the equation x=\frac{5±3\sqrt{5}}{10} when ± is plus. Add 5 to 3\sqrt{5}.

$$x=\frac{3\sqrt{5}+5}{10}$$

Divide 5+3\sqrt{5} by 10.

$$x=\frac{3\sqrt{5}}{10}+\frac{1}{2}$$

Now solve the equation x=\frac{5±3\sqrt{5}}{10} when ± is minus. Subtract 3\sqrt{5} from 5.

$$x=\frac{5-3\sqrt{5}}{10}$$

Divide 5-3\sqrt{5} by 10.

$$x=-\frac{3\sqrt{5}}{10}+\frac{1}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1}{2}+\frac{3\sqrt{5}}{10} for x_{1} and \frac{1}{2}-\frac{3\sqrt{5}}{10} for x_{2}.

$$5x^{2}-5x-1=5\left(x-\left(\frac{3\sqrt{5}}{10}+\frac{1}{2}\right)\right)\left(x-\left(-\frac{3\sqrt{5}}{10}+\frac{1}{2}\right)\right)$$

Evaluate

$$5x^{2}-5x-1$$

Solution Steps

Combine 3x^{2} and 2x^{2} to get 5x^{2}.

$$5x^{2}-5x-1$$