Solve 32x^2-56x+20x-35+15x^2-40

$$32 x ^ { 2 } - 56 x + 20 x - 35 + 15 x ^ { 2 } - 40$$

Evaluate

$$47x^{2}-36x-75$$

Solution Steps

Combine -56x and 20x to get -36x.

$$32x^{2}-36x-35+15x^{2}-40$$

Combine 32x^{2} and 15x^{2} to get 47x^{2}.

$$47x^{2}-36x-35-40$$

Subtract 40 from -35 to get -75.

$$47x^{2}-36x-75$$

Factor

$$47\left(x-\frac{18-\sqrt{3849}}{47}\right)\left(x-\frac{\sqrt{3849}+18}{47}\right)$$

Steps Using the Quadratic Formula

Combine -56x and 20x to get -36x.

$$factor(32x^{2}-36x-35+15x^{2}-40)$$

Combine 32x^{2} and 15x^{2} to get 47x^{2}.

$$factor(47x^{2}-36x-35-40)$$

Subtract 40 from -35 to get -75.

$$factor(47x^{2}-36x-75)$$

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.

$$47x^{2}-36x-75=0$$

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.

$$x=\frac{-\left(-36\right)±\sqrt{\left(-36\right)^{2}-4\times 47\left(-75\right)}}{2\times 47}$$

Square -36.

$$x=\frac{-\left(-36\right)±\sqrt{1296-4\times 47\left(-75\right)}}{2\times 47}$$

Multiply -4 times 47.

$$x=\frac{-\left(-36\right)±\sqrt{1296-188\left(-75\right)}}{2\times 47}$$

Multiply -188 times -75.

$$x=\frac{-\left(-36\right)±\sqrt{1296+14100}}{2\times 47}$$

Add 1296 to 14100.

$$x=\frac{-\left(-36\right)±\sqrt{15396}}{2\times 47}$$

Take the square root of 15396.

$$x=\frac{-\left(-36\right)±2\sqrt{3849}}{2\times 47}$$

The opposite of -36 is 36.

$$x=\frac{36±2\sqrt{3849}}{2\times 47}$$

Multiply 2 times 47.

$$x=\frac{36±2\sqrt{3849}}{94}$$

Now solve the equation x=\frac{36±2\sqrt{3849}}{94} when ± is plus. Add 36 to 2\sqrt{3849}.

$$x=\frac{2\sqrt{3849}+36}{94}$$

Divide 36+2\sqrt{3849} by 94.

$$x=\frac{\sqrt{3849}+18}{47}$$

Now solve the equation x=\frac{36±2\sqrt{3849}}{94} when ± is minus. Subtract 2\sqrt{3849} from 36.

$$x=\frac{36-2\sqrt{3849}}{94}$$

Divide 36-2\sqrt{3849} by 94.

$$x=\frac{18-\sqrt{3849}}{47}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{18+\sqrt{3849}}{47} for x_{1} and \frac{18-\sqrt{3849}}{47} for x_{2}.

$$47x^{2}-36x-75=47\left(x-\frac{\sqrt{3849}+18}{47}\right)\left(x-\frac{18-\sqrt{3849}}{47}\right)$$