Solve -10x^2-x+8x-7+15x^2-9

$$- 10 x ^ { 2 } - x + 8 x - 7 + 15 x ^ { 2 } - 9$$

Evaluate

$$5x^{2}+7x-16$$

Solution Steps

Combine -x and 8x to get 7x.

$$-10x^{2}+7x-7+15x^{2}-9$$

Combine -10x^{2} and 15x^{2} to get 5x^{2}.

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

Subtract 9 from -7 to get -16.

$$5x^{2}+7x-16$$

Factor

$$5\left(x-\frac{-3\sqrt{41}-7}{10}\right)\left(x-\frac{3\sqrt{41}-7}{10}\right)$$

Steps Using the Quadratic Formula

Combine -x and 8x to get 7x.

$$factor(-10x^{2}+7x-7+15x^{2}-9)$$

Combine -10x^{2} and 15x^{2} to get 5x^{2}.

$$factor(5x^{2}+7x-7-9)$$

Subtract 9 from -7 to get -16.

$$factor(5x^{2}+7x-16)$$

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.

$$5x^{2}+7x-16=0$$

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.

$$x=\frac{-7±\sqrt{7^{2}-4\times 5\left(-16\right)}}{2\times 5}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\times 5\left(-16\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-7±\sqrt{49-20\left(-16\right)}}{2\times 5}$$

Multiply -20 times -16.

$$x=\frac{-7±\sqrt{49+320}}{2\times 5}$$

Add 49 to 320.

$$x=\frac{-7±\sqrt{369}}{2\times 5}$$

Take the square root of 369.

$$x=\frac{-7±3\sqrt{41}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-7±3\sqrt{41}}{10}$$

Now solve the equation x=\frac{-7±3\sqrt{41}}{10} when ± is plus. Add -7 to 3\sqrt{41}.

$$x=\frac{3\sqrt{41}-7}{10}$$

Now solve the equation x=\frac{-7±3\sqrt{41}}{10} when ± is minus. Subtract 3\sqrt{41} from -7.

$$x=\frac{-3\sqrt{41}-7}{10}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-7+3\sqrt{41}}{10} for x_{1} and \frac{-7-3\sqrt{41}}{10} for x_{2}.

$$5x^{2}+7x-16=5\left(x-\frac{3\sqrt{41}-7}{10}\right)\left(x-\frac{-3\sqrt{41}-7}{10}\right)$$