Solve (-17x^2-13x-15)+(21x^2+29x-35)

$$( - 17 x ^ { 2 } - 13 x - 15 ) + ( 21 x ^ { 2 } + 29 x - 35 )$$

Evaluate

$$4x^{2}+16x-50$$

Solution Steps

Combine -17x^{2} and 21x^{2} to get 4x^{2}.

$$4x^{2}-13x-15+29x-35$$

Combine -13x and 29x to get 16x.

$$4x^{2}+16x-15-35$$

Subtract 35 from -15 to get -50.

$$4x^{2}+16x-50$$

Factor

$$4\left(x-\left(-\frac{\sqrt{66}}{2}-2\right)\right)\left(x-\left(\frac{\sqrt{66}}{2}-2\right)\right)$$

Steps Using the Quadratic Formula

Combine -17x^{2} and 21x^{2} to get 4x^{2}.

$$factor(4x^{2}-13x-15+29x-35)$$

Combine -13x and 29x to get 16x.

$$factor(4x^{2}+16x-15-35)$$

Subtract 35 from -15 to get -50.

$$factor(4x^{2}+16x-50)$$

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.

$$4x^{2}+16x-50=0$$

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.

$$x=\frac{-16±\sqrt{16^{2}-4\times 4\left(-50\right)}}{2\times 4}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\times 4\left(-50\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-16±\sqrt{256-16\left(-50\right)}}{2\times 4}$$

Multiply -16 times -50.

$$x=\frac{-16±\sqrt{256+800}}{2\times 4}$$

Add 256 to 800.

$$x=\frac{-16±\sqrt{1056}}{2\times 4}$$

Take the square root of 1056.

$$x=\frac{-16±4\sqrt{66}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-16±4\sqrt{66}}{8}$$

Now solve the equation x=\frac{-16±4\sqrt{66}}{8} when ± is plus. Add -16 to 4\sqrt{66}.

$$x=\frac{4\sqrt{66}-16}{8}$$

Divide -16+4\sqrt{66} by 8.

$$x=\frac{\sqrt{66}}{2}-2$$

Now solve the equation x=\frac{-16±4\sqrt{66}}{8} when ± is minus. Subtract 4\sqrt{66} from -16.

$$x=\frac{-4\sqrt{66}-16}{8}$$

Divide -16-4\sqrt{66} by 8.

$$x=-\frac{\sqrt{66}}{2}-2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -2+\frac{\sqrt{66}}{2} for x_{1} and -2-\frac{\sqrt{66}}{2} for x_{2}.

$$4x^{2}+16x-50=4\left(x-\left(\frac{\sqrt{66}}{2}-2\right)\right)\left(x-\left(-\frac{\sqrt{66}}{2}-2\right)\right)$$