Solve (x^2-5x+7)+(2x^2-8x-5)

$$( x ^ { 2 } - 5 x + 7 ) + ( 2 x ^ { 2 } - 8 x - 5 )$$

Evaluate

$$3x^{2}-13x+2$$

Solution Steps

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

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

Combine -5x and -8x to get -13x.

$$3x^{2}-13x+7-5$$

Subtract 5 from 7 to get 2.

$$3x^{2}-13x+2$$

Factor

$$3\left(x-\frac{13-\sqrt{145}}{6}\right)\left(x-\frac{\sqrt{145}+13}{6}\right)$$

Steps Using the Quadratic Formula

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

$$factor(3x^{2}-5x+7-8x-5)$$

Combine -5x and -8x to get -13x.

$$factor(3x^{2}-13x+7-5)$$

Subtract 5 from 7 to get 2.

$$factor(3x^{2}-13x+2)$$

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.

$$3x^{2}-13x+2=0$$

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.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 3\times 2}}{2\times 3}$$

Square -13.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\times 3\times 2}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-13\right)±\sqrt{169-12\times 2}}{2\times 3}$$

Multiply -12 times 2.

$$x=\frac{-\left(-13\right)±\sqrt{169-24}}{2\times 3}$$

Add 169 to -24.

$$x=\frac{-\left(-13\right)±\sqrt{145}}{2\times 3}$$

The opposite of -13 is 13.

$$x=\frac{13±\sqrt{145}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{13±\sqrt{145}}{6}$$

Now solve the equation x=\frac{13±\sqrt{145}}{6} when ± is plus. Add 13 to \sqrt{145}.

$$x=\frac{\sqrt{145}+13}{6}$$

Now solve the equation x=\frac{13±\sqrt{145}}{6} when ± is minus. Subtract \sqrt{145} from 13.

$$x=\frac{13-\sqrt{145}}{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{13+\sqrt{145}}{6} for x_{1} and \frac{13-\sqrt{145}}{6} for x_{2}.

$$3x^{2}-13x+2=3\left(x-\frac{\sqrt{145}+13}{6}\right)\left(x-\frac{13-\sqrt{145}}{6}\right)$$