Solve 9x^2+15x+7-5x-8

$$9 { x }^{ 2 } +15x+7-5x-8$$

Evaluate

$$9x^{2}+10x-1$$

Solution Steps

Combine 15x and -5x to get 10x.

$$9x^{2}+10x+7-8$$

Subtract 8 from 7 to get -1.

$$9x^{2}+10x-1$$

Factor

$$9\left(x-\frac{-\sqrt{34}-5}{9}\right)\left(x-\frac{\sqrt{34}-5}{9}\right)$$

Steps Using the Quadratic Formula

Combine 15x and -5x to get 10x.

$$factor(9x^{2}+10x+7-8)$$

Subtract 8 from 7 to get -1.

$$factor(9x^{2}+10x-1)$$

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.

$$9x^{2}+10x-1=0$$

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.

$$x=\frac{-10±\sqrt{10^{2}-4\times 9\left(-1\right)}}{2\times 9}$$

Square 10.

$$x=\frac{-10±\sqrt{100-4\times 9\left(-1\right)}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-10±\sqrt{100-36\left(-1\right)}}{2\times 9}$$

Multiply -36 times -1.

$$x=\frac{-10±\sqrt{100+36}}{2\times 9}$$

Add 100 to 36.

$$x=\frac{-10±\sqrt{136}}{2\times 9}$$

Take the square root of 136.

$$x=\frac{-10±2\sqrt{34}}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{-10±2\sqrt{34}}{18}$$

Now solve the equation x=\frac{-10±2\sqrt{34}}{18} when ± is plus. Add -10 to 2\sqrt{34}.

$$x=\frac{2\sqrt{34}-10}{18}$$

Divide -10+2\sqrt{34} by 18.

$$x=\frac{\sqrt{34}-5}{9}$$

Now solve the equation x=\frac{-10±2\sqrt{34}}{18} when ± is minus. Subtract 2\sqrt{34} from -10.

$$x=\frac{-2\sqrt{34}-10}{18}$$

Divide -10-2\sqrt{34} by 18.

$$x=\frac{-\sqrt{34}-5}{9}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-5+\sqrt{34}}{9} for x_{1} and \frac{-5-\sqrt{34}}{9} for x_{2}.

$$9x^{2}+10x-1=9\left(x-\frac{\sqrt{34}-5}{9}\right)\left(x-\frac{-\sqrt{34}-5}{9}\right)$$