Solve 5x^2-7x+x-3

$$5 x ^ { 2 } - 7 x + x - 3$$

Factor

$$5\left(x-\frac{3-2\sqrt{6}}{5}\right)\left(x-\frac{2\sqrt{6}+3}{5}\right)$$

Steps Using the Quadratic Formula

Combine -7x and x to get -6x.

$$factor(5x^{2}-6x-3)$$

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.

$$5x^{2}-6x-3=0$$

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.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 5\left(-3\right)}}{2\times 5}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 5\left(-3\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-6\right)±\sqrt{36-20\left(-3\right)}}{2\times 5}$$

Multiply -20 times -3.

$$x=\frac{-\left(-6\right)±\sqrt{36+60}}{2\times 5}$$

Add 36 to 60.

$$x=\frac{-\left(-6\right)±\sqrt{96}}{2\times 5}$$

Take the square root of 96.

$$x=\frac{-\left(-6\right)±4\sqrt{6}}{2\times 5}$$

The opposite of -6 is 6.

$$x=\frac{6±4\sqrt{6}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{6±4\sqrt{6}}{10}$$

Now solve the equation x=\frac{6±4\sqrt{6}}{10} when ± is plus. Add 6 to 4\sqrt{6}.

$$x=\frac{4\sqrt{6}+6}{10}$$

Divide 6+4\sqrt{6} by 10.

$$x=\frac{2\sqrt{6}+3}{5}$$

Now solve the equation x=\frac{6±4\sqrt{6}}{10} when ± is minus. Subtract 4\sqrt{6} from 6.

$$x=\frac{6-4\sqrt{6}}{10}$$

Divide 6-4\sqrt{6} by 10.

$$x=\frac{3-2\sqrt{6}}{5}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{3+2\sqrt{6}}{5} for x_{1} and \frac{3-2\sqrt{6}}{5} for x_{2}.

$$5x^{2}-6x-3=5\left(x-\frac{2\sqrt{6}+3}{5}\right)\left(x-\frac{3-2\sqrt{6}}{5}\right)$$

Evaluate

$$5x^{2}-6x-3$$

Solution Steps

Combine -7x and x to get -6x.

$$5x^{2}-6x-3$$