Solve (8x^2-5x+2)+(9x^2+4x-3)

$$( 8 x ^ { 2 } - 5 x + 2 ) + ( 9 x ^ { 2 } + 4 x - 3 )$$

Evaluate

$$17x^{2}-x-1$$

Solution Steps

Combine 8x^{2} and 9x^{2} to get 17x^{2}.

$$17x^{2}-5x+2+4x-3$$

Combine -5x and 4x to get -x.

$$17x^{2}-x+2-3$$

Subtract 3 from 2 to get -1.

$$17x^{2}-x-1$$

Factor

$$17\left(x-\frac{1-\sqrt{69}}{34}\right)\left(x-\frac{\sqrt{69}+1}{34}\right)$$

Steps Using the Quadratic Formula

Combine 8x^{2} and 9x^{2} to get 17x^{2}.

$$factor(17x^{2}-5x+2+4x-3)$$

Combine -5x and 4x to get -x.

$$factor(17x^{2}-x+2-3)$$

Subtract 3 from 2 to get -1.

$$factor(17x^{2}-x-1)$$

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.

$$17x^{2}-x-1=0$$

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.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 17\left(-1\right)}}{2\times 17}$$

Multiply -4 times 17.

$$x=\frac{-\left(-1\right)±\sqrt{1-68\left(-1\right)}}{2\times 17}$$

Multiply -68 times -1.

$$x=\frac{-\left(-1\right)±\sqrt{1+68}}{2\times 17}$$

Add 1 to 68.

$$x=\frac{-\left(-1\right)±\sqrt{69}}{2\times 17}$$

The opposite of -1 is 1.

$$x=\frac{1±\sqrt{69}}{2\times 17}$$

Multiply 2 times 17.

$$x=\frac{1±\sqrt{69}}{34}$$

Now solve the equation x=\frac{1±\sqrt{69}}{34} when ± is plus. Add 1 to \sqrt{69}.

$$x=\frac{\sqrt{69}+1}{34}$$

Now solve the equation x=\frac{1±\sqrt{69}}{34} when ± is minus. Subtract \sqrt{69} from 1.

$$x=\frac{1-\sqrt{69}}{34}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1+\sqrt{69}}{34} for x_{1} and \frac{1-\sqrt{69}}{34} for x_{2}.

$$17x^{2}-x-1=17\left(x-\frac{\sqrt{69}+1}{34}\right)\left(x-\frac{1-\sqrt{69}}{34}\right)$$