Solve 10x^2+4-6x-8x^2-15x

$$10 x ^ { 2 } + 4 - 6 x - 8 x ^ { 2 } - 15 x$$

Evaluate

$$2x^{2}-21x+4$$

Solution Steps

Combine 10x^{2} and -8x^{2} to get 2x^{2}.

$$2x^{2}+4-6x-15x$$

Combine -6x and -15x to get -21x.

$$2x^{2}+4-21x$$

Factor

$$2\left(x-\frac{21-\sqrt{409}}{4}\right)\left(x-\frac{\sqrt{409}+21}{4}\right)$$

Steps Using the Quadratic Formula

Combine 10x^{2} and -8x^{2} to get 2x^{2}.

$$factor(2x^{2}+4-6x-15x)$$

Combine -6x and -15x to get -21x.

$$factor(2x^{2}+4-21x)$$

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.

$$2x^{2}-21x+4=0$$

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.

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

Square -21.

$$x=\frac{-\left(-21\right)±\sqrt{441-4\times 2\times 4}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-21\right)±\sqrt{441-8\times 4}}{2\times 2}$$

Multiply -8 times 4.

$$x=\frac{-\left(-21\right)±\sqrt{441-32}}{2\times 2}$$

Add 441 to -32.

$$x=\frac{-\left(-21\right)±\sqrt{409}}{2\times 2}$$

The opposite of -21 is 21.

$$x=\frac{21±\sqrt{409}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{21±\sqrt{409}}{4}$$

Now solve the equation x=\frac{21±\sqrt{409}}{4} when ± is plus. Add 21 to \sqrt{409}.

$$x=\frac{\sqrt{409}+21}{4}$$

Now solve the equation x=\frac{21±\sqrt{409}}{4} when ± is minus. Subtract \sqrt{409} from 21.

$$x=\frac{21-\sqrt{409}}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{21+\sqrt{409}}{4} for x_{1} and \frac{21-\sqrt{409}}{4} for x_{2}.

$$2x^{2}-21x+4=2\left(x-\frac{\sqrt{409}+21}{4}\right)\left(x-\frac{21-\sqrt{409}}{4}\right)$$