Solve 12x^2+8x-3-7x^2+4x+5

$$12 x ^ { 2 } + 8 x - 3 - 7 x ^ { 2 } + 4 x + 5$$

Evaluate

$$5x^{2}+12x+2$$

Solution Steps

Combine 12x^{2} and -7x^{2} to get 5x^{2}.

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

Combine 8x and 4x to get 12x.

$$5x^{2}+12x-3+5$$

Add -3 and 5 to get 2.

$$5x^{2}+12x+2$$

Factor

$$5\left(x-\frac{-\sqrt{26}-6}{5}\right)\left(x-\frac{\sqrt{26}-6}{5}\right)$$

Steps Using the Quadratic Formula

Combine 12x^{2} and -7x^{2} to get 5x^{2}.

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

Combine 8x and 4x to get 12x.

$$factor(5x^{2}+12x-3+5)$$

Add -3 and 5 to get 2.

$$factor(5x^{2}+12x+2)$$

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.

$$5x^{2}+12x+2=0$$

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.

$$x=\frac{-12±\sqrt{12^{2}-4\times 5\times 2}}{2\times 5}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\times 5\times 2}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-12±\sqrt{144-20\times 2}}{2\times 5}$$

Multiply -20 times 2.

$$x=\frac{-12±\sqrt{144-40}}{2\times 5}$$

Add 144 to -40.

$$x=\frac{-12±\sqrt{104}}{2\times 5}$$

Take the square root of 104.

$$x=\frac{-12±2\sqrt{26}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-12±2\sqrt{26}}{10}$$

Now solve the equation x=\frac{-12±2\sqrt{26}}{10} when ± is plus. Add -12 to 2\sqrt{26}.

$$x=\frac{2\sqrt{26}-12}{10}$$

Divide -12+2\sqrt{26} by 10.

$$x=\frac{\sqrt{26}-6}{5}$$

Now solve the equation x=\frac{-12±2\sqrt{26}}{10} when ± is minus. Subtract 2\sqrt{26} from -12.

$$x=\frac{-2\sqrt{26}-12}{10}$$

Divide -12-2\sqrt{26} by 10.

$$x=\frac{-\sqrt{26}-6}{5}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-6+\sqrt{26}}{5} for x_{1} and \frac{-6-\sqrt{26}}{5} for x_{2}.

$$5x^{2}+12x+2=5\left(x-\frac{\sqrt{26}-6}{5}\right)\left(x-\frac{-\sqrt{26}-6}{5}\right)$$