Solve (3x^2-3x+28)+(7x^2+9x-89)

$$( 3 x ^ { 2 } - 3 x + 28 ) + ( 7 x ^ { 2 } + 9 x - 89 )$$

Evaluate

$$10x^{2}+6x-61$$

Solution Steps

Combine 3x^{2} and 7x^{2} to get 10x^{2}.

$$10x^{2}-3x+28+9x-89$$

Combine -3x and 9x to get 6x.

$$10x^{2}+6x+28-89$$

Subtract 89 from 28 to get -61.

$$10x^{2}+6x-61$$

Factor

$$10\left(x-\frac{-\sqrt{619}-3}{10}\right)\left(x-\frac{\sqrt{619}-3}{10}\right)$$

Steps Using the Quadratic Formula

Combine 3x^{2} and 7x^{2} to get 10x^{2}.

$$factor(10x^{2}-3x+28+9x-89)$$

Combine -3x and 9x to get 6x.

$$factor(10x^{2}+6x+28-89)$$

Subtract 89 from 28 to get -61.

$$factor(10x^{2}+6x-61)$$

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.

$$10x^{2}+6x-61=0$$

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.

$$x=\frac{-6±\sqrt{6^{2}-4\times 10\left(-61\right)}}{2\times 10}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\times 10\left(-61\right)}}{2\times 10}$$

Multiply -4 times 10.

$$x=\frac{-6±\sqrt{36-40\left(-61\right)}}{2\times 10}$$

Multiply -40 times -61.

$$x=\frac{-6±\sqrt{36+2440}}{2\times 10}$$

Add 36 to 2440.

$$x=\frac{-6±\sqrt{2476}}{2\times 10}$$

Take the square root of 2476.

$$x=\frac{-6±2\sqrt{619}}{2\times 10}$$

Multiply 2 times 10.

$$x=\frac{-6±2\sqrt{619}}{20}$$

Now solve the equation x=\frac{-6±2\sqrt{619}}{20} when ± is plus. Add -6 to 2\sqrt{619}.

$$x=\frac{2\sqrt{619}-6}{20}$$

Divide -6+2\sqrt{619} by 20.

$$x=\frac{\sqrt{619}-3}{10}$$

Now solve the equation x=\frac{-6±2\sqrt{619}}{20} when ± is minus. Subtract 2\sqrt{619} from -6.

$$x=\frac{-2\sqrt{619}-6}{20}$$

Divide -6-2\sqrt{619} by 20.

$$x=\frac{-\sqrt{619}-3}{10}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-3+\sqrt{619}}{10} for x_{1} and \frac{-3-\sqrt{619}}{10} for x_{2}.

$$10x^{2}+6x-61=10\left(x-\frac{\sqrt{619}-3}{10}\right)\left(x-\frac{-\sqrt{619}-3}{10}\right)$$