Solve (4x^2-8x+3)+(x^2-5)

$$( 4 x ^ { 2 } - 8 x + 3 ) + ( x ^ { 2 } - 5 )$$

Evaluate

$$5x^{2}-8x-2$$

Solution Steps

Combine 4x^{2} and x^{2} to get 5x^{2}.

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

Subtract 5 from 3 to get -2.

$$5x^{2}-8x-2$$

Factor

$$5\left(x-\frac{4-\sqrt{26}}{5}\right)\left(x-\frac{\sqrt{26}+4}{5}\right)$$

Steps Using the Quadratic Formula

Combine 4x^{2} and x^{2} to get 5x^{2}.

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

Subtract 5 from 3 to get -2.

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

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.

$$5x^{2}-8x-2=0$$

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.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 5\left(-2\right)}}{2\times 5}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 5\left(-2\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-8\right)±\sqrt{64-20\left(-2\right)}}{2\times 5}$$

Multiply -20 times -2.

$$x=\frac{-\left(-8\right)±\sqrt{64+40}}{2\times 5}$$

Add 64 to 40.

$$x=\frac{-\left(-8\right)±\sqrt{104}}{2\times 5}$$

Take the square root of 104.

$$x=\frac{-\left(-8\right)±2\sqrt{26}}{2\times 5}$$

The opposite of -8 is 8.

$$x=\frac{8±2\sqrt{26}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{8±2\sqrt{26}}{10}$$

Now solve the equation x=\frac{8±2\sqrt{26}}{10} when ± is plus. Add 8 to 2\sqrt{26}.

$$x=\frac{2\sqrt{26}+8}{10}$$

Divide 8+2\sqrt{26} by 10.

$$x=\frac{\sqrt{26}+4}{5}$$

Now solve the equation x=\frac{8±2\sqrt{26}}{10} when ± is minus. Subtract 2\sqrt{26} from 8.

$$x=\frac{8-2\sqrt{26}}{10}$$

Divide 8-2\sqrt{26} by 10.

$$x=\frac{4-\sqrt{26}}{5}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{4+\sqrt{26}}{5} for x_{1} and \frac{4-\sqrt{26}}{5} for x_{2}.

$$5x^{2}-8x-2=5\left(x-\frac{\sqrt{26}+4}{5}\right)\left(x-\frac{4-\sqrt{26}}{5}\right)$$