Solve (5x^2+7x-2)+(7x^2+4x-4)

$$( 5 x ^ { 2 } + 7 x - 2 ) + ( 7 x ^ { 2 } + 4 x - 4 )$$

Evaluate

$$12x^{2}+11x-6$$

Solution Steps

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

$$12x^{2}+7x-2+4x-4$$

Combine 7x and 4x to get 11x.

$$12x^{2}+11x-2-4$$

Subtract 4 from -2 to get -6.

$$12x^{2}+11x-6$$

Factor

$$12\left(x-\frac{-\sqrt{409}-11}{24}\right)\left(x-\frac{\sqrt{409}-11}{24}\right)$$

Steps Using the Quadratic Formula

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

$$factor(12x^{2}+7x-2+4x-4)$$

Combine 7x and 4x to get 11x.

$$factor(12x^{2}+11x-2-4)$$

Subtract 4 from -2 to get -6.

$$factor(12x^{2}+11x-6)$$

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.

$$12x^{2}+11x-6=0$$

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.

$$x=\frac{-11±\sqrt{11^{2}-4\times 12\left(-6\right)}}{2\times 12}$$

Square 11.

$$x=\frac{-11±\sqrt{121-4\times 12\left(-6\right)}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-11±\sqrt{121-48\left(-6\right)}}{2\times 12}$$

Multiply -48 times -6.

$$x=\frac{-11±\sqrt{121+288}}{2\times 12}$$

Add 121 to 288.

$$x=\frac{-11±\sqrt{409}}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{-11±\sqrt{409}}{24}$$

Now solve the equation x=\frac{-11±\sqrt{409}}{24} when ± is plus. Add -11 to \sqrt{409}.

$$x=\frac{\sqrt{409}-11}{24}$$

Now solve the equation x=\frac{-11±\sqrt{409}}{24} when ± is minus. Subtract \sqrt{409} from -11.

$$x=\frac{-\sqrt{409}-11}{24}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-11+\sqrt{409}}{24} for x_{1} and \frac{-11-\sqrt{409}}{24} for x_{2}.

$$12x^{2}+11x-6=12\left(x-\frac{\sqrt{409}-11}{24}\right)\left(x-\frac{-\sqrt{409}-11}{24}\right)$$