Solve (-7x^2-2x+3)+(4x^2-9x)

$$( - 7 x ^ { 2 } - 2 x + 3 ) + ( 4 x ^ { 2 } - 9 x )$$

Evaluate

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

Solution Steps

Combine -7x^{2} and 4x^{2} to get -3x^{2}.

$$-3x^{2}-2x+3-9x$$

Combine -2x and -9x to get -11x.

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

Factor

$$-3\left(x-\frac{-\sqrt{157}-11}{6}\right)\left(x-\frac{\sqrt{157}-11}{6}\right)$$

Steps Using the Quadratic Formula

Combine -7x^{2} and 4x^{2} to get -3x^{2}.

$$factor(-3x^{2}-2x+3-9x)$$

Combine -2x and -9x to get -11x.

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

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.

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

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.

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

Square -11.

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

Multiply -4 times -3.

$$x=\frac{-\left(-11\right)±\sqrt{121+12\times 3}}{2\left(-3\right)}$$

Multiply 12 times 3.

$$x=\frac{-\left(-11\right)±\sqrt{121+36}}{2\left(-3\right)}$$

Add 121 to 36.

$$x=\frac{-\left(-11\right)±\sqrt{157}}{2\left(-3\right)}$$

The opposite of -11 is 11.

$$x=\frac{11±\sqrt{157}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{11±\sqrt{157}}{-6}$$

Now solve the equation x=\frac{11±\sqrt{157}}{-6} when ± is plus. Add 11 to \sqrt{157}.

$$x=\frac{\sqrt{157}+11}{-6}$$

Divide 11+\sqrt{157} by -6.

$$x=\frac{-\sqrt{157}-11}{6}$$

Now solve the equation x=\frac{11±\sqrt{157}}{-6} when ± is minus. Subtract \sqrt{157} from 11.

$$x=\frac{11-\sqrt{157}}{-6}$$

Divide 11-\sqrt{157} by -6.

$$x=\frac{\sqrt{157}-11}{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-11-\sqrt{157}}{6} for x_{1} and \frac{-11+\sqrt{157}}{6} for x_{2}.

$$-3x^{2}-11x+3=-3\left(x-\frac{-\sqrt{157}-11}{6}\right)\left(x-\frac{\sqrt{157}-11}{6}\right)$$