Solve 10x^2-8x+3-34x^2+19x

$$10 x ^ { 2 } - 8 x + 3 - 34 x ^ { 2 } + 19 x$$

Evaluate

$$3+11x-24x^{2}$$

Solution Steps

Combine 10x^{2} and -34x^{2} to get -24x^{2}.

$$-24x^{2}-8x+3+19x$$

Combine -8x and 19x to get 11x.

$$-24x^{2}+11x+3$$

Factor

$$-24\left(x-\frac{11-\sqrt{409}}{48}\right)\left(x-\frac{\sqrt{409}+11}{48}\right)$$

Steps Using the Quadratic Formula

Combine 10x^{2} and -34x^{2} to get -24x^{2}.

$$factor(-24x^{2}-8x+3+19x)$$

Combine -8x and 19x to get 11x.

$$factor(-24x^{2}+11x+3)$$

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.

$$-24x^{2}+11x+3=0$$

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.

$$x=\frac{-11±\sqrt{11^{2}-4\left(-24\right)\times 3}}{2\left(-24\right)}$$

Square 11.

$$x=\frac{-11±\sqrt{121-4\left(-24\right)\times 3}}{2\left(-24\right)}$$

Multiply -4 times -24.

$$x=\frac{-11±\sqrt{121+96\times 3}}{2\left(-24\right)}$$

Multiply 96 times 3.

$$x=\frac{-11±\sqrt{121+288}}{2\left(-24\right)}$$

Add 121 to 288.

$$x=\frac{-11±\sqrt{409}}{2\left(-24\right)}$$

Multiply 2 times -24.

$$x=\frac{-11±\sqrt{409}}{-48}$$

Now solve the equation x=\frac{-11±\sqrt{409}}{-48} when ± is plus. Add -11 to \sqrt{409}.

$$x=\frac{\sqrt{409}-11}{-48}$$

Divide -11+\sqrt{409} by -48.

$$x=\frac{11-\sqrt{409}}{48}$$

Now solve the equation x=\frac{-11±\sqrt{409}}{-48} when ± is minus. Subtract \sqrt{409} from -11.

$$x=\frac{-\sqrt{409}-11}{-48}$$

Divide -11-\sqrt{409} by -48.

$$x=\frac{\sqrt{409}+11}{48}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{11-\sqrt{409}}{48} for x_{1} and \frac{11+\sqrt{409}}{48} for x_{2}.

$$-24x^{2}+11x+3=-24\left(x-\frac{11-\sqrt{409}}{48}\right)\left(x-\frac{\sqrt{409}+11}{48}\right)$$