Solve (-9x^2+7x-5)+(-5x^2+9)

$$( - 9 x ^ { 2 } + 7 x - 5 ) + ( - 5 x ^ { 2 } + 9 )$$

Evaluate

$$4+7x-14x^{2}$$

Solution Steps

Combine -9x^{2} and -5x^{2} to get -14x^{2}.

$$-14x^{2}+7x-5+9$$

Add -5 and 9 to get 4.

$$-14x^{2}+7x+4$$

Factor

$$-14\left(x-\left(-\frac{\sqrt{273}}{28}+\frac{1}{4}\right)\right)\left(x-\left(\frac{\sqrt{273}}{28}+\frac{1}{4}\right)\right)$$

Steps Using the Quadratic Formula

Combine -9x^{2} and -5x^{2} to get -14x^{2}.

$$factor(-14x^{2}+7x-5+9)$$

Add -5 and 9 to get 4.

$$factor(-14x^{2}+7x+4)$$

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.

$$-14x^{2}+7x+4=0$$

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.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-14\right)\times 4}}{2\left(-14\right)}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\left(-14\right)\times 4}}{2\left(-14\right)}$$

Multiply -4 times -14.

$$x=\frac{-7±\sqrt{49+56\times 4}}{2\left(-14\right)}$$

Multiply 56 times 4.

$$x=\frac{-7±\sqrt{49+224}}{2\left(-14\right)}$$

Add 49 to 224.

$$x=\frac{-7±\sqrt{273}}{2\left(-14\right)}$$

Multiply 2 times -14.

$$x=\frac{-7±\sqrt{273}}{-28}$$

Now solve the equation x=\frac{-7±\sqrt{273}}{-28} when ± is plus. Add -7 to \sqrt{273}.

$$x=\frac{\sqrt{273}-7}{-28}$$

Divide -7+\sqrt{273} by -28.

$$x=-\frac{\sqrt{273}}{28}+\frac{1}{4}$$

Now solve the equation x=\frac{-7±\sqrt{273}}{-28} when ± is minus. Subtract \sqrt{273} from -7.

$$x=\frac{-\sqrt{273}-7}{-28}$$

Divide -7-\sqrt{273} by -28.

$$x=\frac{\sqrt{273}}{28}+\frac{1}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1}{4}-\frac{\sqrt{273}}{28} for x_{1} and \frac{1}{4}+\frac{\sqrt{273}}{28} for x_{2}.

$$-14x^{2}+7x+4=-14\left(x-\left(-\frac{\sqrt{273}}{28}+\frac{1}{4}\right)\right)\left(x-\left(\frac{\sqrt{273}}{28}+\frac{1}{4}\right)\right)$$