Solve 9x^2-15x+4quad-3

$$9 x ^ { 2 } - 15 x + 4 \quad - 3$$

Factor

$$9\left(x-\frac{5-\sqrt{21}}{6}\right)\left(x-\frac{\sqrt{21}+5}{6}\right)$$

Steps Using the Quadratic Formula

Subtract 3 from 4 to get 1.

$$factor(9x^{2}-15x+1)$$

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.

$$9x^{2}-15x+1=0$$

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.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times 9}}{2\times 9}$$

Square -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\times 9}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-\left(-15\right)±\sqrt{225-36}}{2\times 9}$$

Add 225 to -36.

$$x=\frac{-\left(-15\right)±\sqrt{189}}{2\times 9}$$

Take the square root of 189.

$$x=\frac{-\left(-15\right)±3\sqrt{21}}{2\times 9}$$

The opposite of -15 is 15.

$$x=\frac{15±3\sqrt{21}}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{15±3\sqrt{21}}{18}$$

Now solve the equation x=\frac{15±3\sqrt{21}}{18} when ± is plus. Add 15 to 3\sqrt{21}.

$$x=\frac{3\sqrt{21}+15}{18}$$

Divide 15+3\sqrt{21} by 18.

$$x=\frac{\sqrt{21}+5}{6}$$

Now solve the equation x=\frac{15±3\sqrt{21}}{18} when ± is minus. Subtract 3\sqrt{21} from 15.

$$x=\frac{15-3\sqrt{21}}{18}$$

Divide 15-3\sqrt{21} by 18.

$$x=\frac{5-\sqrt{21}}{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{5+\sqrt{21}}{6} for x_{1} and \frac{5-\sqrt{21}}{6} for x_{2}.

$$9x^{2}-15x+1=9\left(x-\frac{\sqrt{21}+5}{6}\right)\left(x-\frac{5-\sqrt{21}}{6}\right)$$

Evaluate

$$9x^{2}-15x+1$$

Solution Steps

Subtract 3 from 4 to get 1.

$$9x^{2}-15x+1$$