Solve (2x^2-7x-15)+(3x^2-6)

$$( 2 x ^ { 2 } - 7 x - 15 ) + ( 3 x ^ { 2 } - 6 )$$

Evaluate

$$5x^{2}-7x-21$$

Solution Steps

Combine 2x^{2} and 3x^{2} to get 5x^{2}.

$$5x^{2}-7x-15-6$$

Subtract 6 from -15 to get -21.

$$5x^{2}-7x-21$$

Factor

$$5\left(x-\frac{7-\sqrt{469}}{10}\right)\left(x-\frac{\sqrt{469}+7}{10}\right)$$

Steps Using the Quadratic Formula

Combine 2x^{2} and 3x^{2} to get 5x^{2}.

$$factor(5x^{2}-7x-15-6)$$

Subtract 6 from -15 to get -21.

$$factor(5x^{2}-7x-21)$$

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.

$$5x^{2}-7x-21=0$$

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.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 5\left(-21\right)}}{2\times 5}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 5\left(-21\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-7\right)±\sqrt{49-20\left(-21\right)}}{2\times 5}$$

Multiply -20 times -21.

$$x=\frac{-\left(-7\right)±\sqrt{49+420}}{2\times 5}$$

Add 49 to 420.

$$x=\frac{-\left(-7\right)±\sqrt{469}}{2\times 5}$$

The opposite of -7 is 7.

$$x=\frac{7±\sqrt{469}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{7±\sqrt{469}}{10}$$

Now solve the equation x=\frac{7±\sqrt{469}}{10} when ± is plus. Add 7 to \sqrt{469}.

$$x=\frac{\sqrt{469}+7}{10}$$

Now solve the equation x=\frac{7±\sqrt{469}}{10} when ± is minus. Subtract \sqrt{469} from 7.

$$x=\frac{7-\sqrt{469}}{10}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7+\sqrt{469}}{10} for x_{1} and \frac{7-\sqrt{469}}{10} for x_{2}.

$$5x^{2}-7x-21=5\left(x-\frac{\sqrt{469}+7}{10}\right)\left(x-\frac{7-\sqrt{469}}{10}\right)$$