Solve (6x^2-4x+7)+(2x^2-3x-9)=

$$( 6 x ^ { 2 } - 4 x + 7 ) + ( 2 x ^ { 2 } - 3 x - 9 ) =$$

Evaluate

$$8x^{2}-7x-2$$

Solution Steps

Combine 6x^{2} and 2x^{2} to get 8x^{2}.

$$8x^{2}-4x+7-3x-9$$

Combine -4x and -3x to get -7x.

$$8x^{2}-7x+7-9$$

Subtract 9 from 7 to get -2.

$$8x^{2}-7x-2$$

Factor

$$8\left(x-\frac{7-\sqrt{113}}{16}\right)\left(x-\frac{\sqrt{113}+7}{16}\right)$$

Steps Using the Quadratic Formula

Combine 6x^{2} and 2x^{2} to get 8x^{2}.

$$factor(8x^{2}-4x+7-3x-9)$$

Combine -4x and -3x to get -7x.

$$factor(8x^{2}-7x+7-9)$$

Subtract 9 from 7 to get -2.

$$factor(8x^{2}-7x-2)$$

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.

$$8x^{2}-7x-2=0$$

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.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 8\left(-2\right)}}{2\times 8}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 8\left(-2\right)}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-\left(-7\right)±\sqrt{49-32\left(-2\right)}}{2\times 8}$$

Multiply -32 times -2.

$$x=\frac{-\left(-7\right)±\sqrt{49+64}}{2\times 8}$$

Add 49 to 64.

$$x=\frac{-\left(-7\right)±\sqrt{113}}{2\times 8}$$

The opposite of -7 is 7.

$$x=\frac{7±\sqrt{113}}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{7±\sqrt{113}}{16}$$

Now solve the equation x=\frac{7±\sqrt{113}}{16} when ± is plus. Add 7 to \sqrt{113}.

$$x=\frac{\sqrt{113}+7}{16}$$

Now solve the equation x=\frac{7±\sqrt{113}}{16} when ± is minus. Subtract \sqrt{113} from 7.

$$x=\frac{7-\sqrt{113}}{16}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7+\sqrt{113}}{16} for x_{1} and \frac{7-\sqrt{113}}{16} for x_{2}.

$$8x^{2}-7x-2=8\left(x-\frac{\sqrt{113}+7}{16}\right)\left(x-\frac{7-\sqrt{113}}{16}\right)$$