Solve 3x^2-x+15+41x^2-14x-25

$$3 { x }^{ 2 } -x+15+41 { x }^{ 2 } -14x-25$$

Evaluate

$$44x^{2}-15x-10$$

Solution Steps

Combine 3x^{2} and 41x^{2} to get 44x^{2}.

$$44x^{2}-x+15-14x-25$$

Combine -x and -14x to get -15x.

$$44x^{2}-15x+15-25$$

Subtract 25 from 15 to get -10.

$$44x^{2}-15x-10$$

Factor

$$44\left(x-\frac{15-\sqrt{1985}}{88}\right)\left(x-\frac{\sqrt{1985}+15}{88}\right)$$

Steps Using the Quadratic Formula

Combine 3x^{2} and 41x^{2} to get 44x^{2}.

$$factor(44x^{2}-x+15-14x-25)$$

Combine -x and -14x to get -15x.

$$factor(44x^{2}-15x+15-25)$$

Subtract 25 from 15 to get -10.

$$factor(44x^{2}-15x-10)$$

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.

$$44x^{2}-15x-10=0$$

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.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times 44\left(-10\right)}}{2\times 44}$$

Square -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\times 44\left(-10\right)}}{2\times 44}$$

Multiply -4 times 44.

$$x=\frac{-\left(-15\right)±\sqrt{225-176\left(-10\right)}}{2\times 44}$$

Multiply -176 times -10.

$$x=\frac{-\left(-15\right)±\sqrt{225+1760}}{2\times 44}$$

Add 225 to 1760.

$$x=\frac{-\left(-15\right)±\sqrt{1985}}{2\times 44}$$

The opposite of -15 is 15.

$$x=\frac{15±\sqrt{1985}}{2\times 44}$$

Multiply 2 times 44.

$$x=\frac{15±\sqrt{1985}}{88}$$

Now solve the equation x=\frac{15±\sqrt{1985}}{88} when ± is plus. Add 15 to \sqrt{1985}.

$$x=\frac{\sqrt{1985}+15}{88}$$

Now solve the equation x=\frac{15±\sqrt{1985}}{88} when ± is minus. Subtract \sqrt{1985} from 15.

$$x=\frac{15-\sqrt{1985}}{88}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{15+\sqrt{1985}}{88} for x_{1} and \frac{15-\sqrt{1985}}{88} for x_{2}.

$$44x^{2}-15x-10=44\left(x-\frac{\sqrt{1985}+15}{88}\right)\left(x-\frac{15-\sqrt{1985}}{88}\right)$$