Solve 6x^2-5-9x^2+14+7x

$$6 x ^ { 2 } - 5 - 9 x ^ { 2 } + 14 + 7 x$$

Evaluate

$$9+7x-3x^{2}$$

Solution Steps

Combine 6x^{2} and -9x^{2} to get -3x^{2}.

$$-3x^{2}-5+14+7x$$

Add -5 and 14 to get 9.

$$-3x^{2}+9+7x$$

Factor

$$-3\left(x-\frac{7-\sqrt{157}}{6}\right)\left(x-\frac{\sqrt{157}+7}{6}\right)$$

Steps Using the Quadratic Formula

Combine 6x^{2} and -9x^{2} to get -3x^{2}.

$$factor(-3x^{2}-5+14+7x)$$

Add -5 and 14 to get 9.

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

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.

$$-3x^{2}+7x+9=0$$

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.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-3\right)\times 9}}{2\left(-3\right)}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\left(-3\right)\times 9}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-7±\sqrt{49+12\times 9}}{2\left(-3\right)}$$

Multiply 12 times 9.

$$x=\frac{-7±\sqrt{49+108}}{2\left(-3\right)}$$

Add 49 to 108.

$$x=\frac{-7±\sqrt{157}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-7±\sqrt{157}}{-6}$$

Now solve the equation x=\frac{-7±\sqrt{157}}{-6} when ± is plus. Add -7 to \sqrt{157}.

$$x=\frac{\sqrt{157}-7}{-6}$$

Divide -7+\sqrt{157} by -6.

$$x=\frac{7-\sqrt{157}}{6}$$

Now solve the equation x=\frac{-7±\sqrt{157}}{-6} when ± is minus. Subtract \sqrt{157} from -7.

$$x=\frac{-\sqrt{157}-7}{-6}$$

Divide -7-\sqrt{157} by -6.

$$x=\frac{\sqrt{157}+7}{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7-\sqrt{157}}{6} for x_{1} and \frac{7+\sqrt{157}}{6} for x_{2}.

$$-3x^{2}+7x+9=-3\left(x-\frac{7-\sqrt{157}}{6}\right)\left(x-\frac{\sqrt{157}+7}{6}\right)$$