Solve (7q^2+q)+(9q+2)

$$( 7 q ^ { 2 } + q ) + ( 9 q + 2 )$$

Factor

$$7\left(q-\frac{-\sqrt{11}-5}{7}\right)\left(q-\frac{\sqrt{11}-5}{7}\right)$$

Steps Using the Quadratic Formula

Combine q and 9q to get 10q.

$$factor(7q^{2}+10q+2)$$

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.

$$7q^{2}+10q+2=0$$

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.

$$q=\frac{-10±\sqrt{10^{2}-4\times 7\times 2}}{2\times 7}$$

Square 10.

$$q=\frac{-10±\sqrt{100-4\times 7\times 2}}{2\times 7}$$

Multiply -4 times 7.

$$q=\frac{-10±\sqrt{100-28\times 2}}{2\times 7}$$

Multiply -28 times 2.

$$q=\frac{-10±\sqrt{100-56}}{2\times 7}$$

Add 100 to -56.

$$q=\frac{-10±\sqrt{44}}{2\times 7}$$

Take the square root of 44.

$$q=\frac{-10±2\sqrt{11}}{2\times 7}$$

Multiply 2 times 7.

$$q=\frac{-10±2\sqrt{11}}{14}$$

Now solve the equation q=\frac{-10±2\sqrt{11}}{14} when ± is plus. Add -10 to 2\sqrt{11}.

$$q=\frac{2\sqrt{11}-10}{14}$$

Divide -10+2\sqrt{11} by 14.

$$q=\frac{\sqrt{11}-5}{7}$$

Now solve the equation q=\frac{-10±2\sqrt{11}}{14} when ± is minus. Subtract 2\sqrt{11} from -10.

$$q=\frac{-2\sqrt{11}-10}{14}$$

Divide -10-2\sqrt{11} by 14.

$$q=\frac{-\sqrt{11}-5}{7}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-5+\sqrt{11}}{7} for x_{1} and \frac{-5-\sqrt{11}}{7} for x_{2}.

$$7q^{2}+10q+2=7\left(q-\frac{\sqrt{11}-5}{7}\right)\left(q-\frac{-\sqrt{11}-5}{7}\right)$$

Evaluate

$$7q^{2}+10q+2$$

Solution Steps

Combine q and 9q to get 10q.

$$7q^{2}+10q+2$$