Solve 7+7x-3x^2-8

$$7 + 7 x - 3 x ^ { 2 } - 8$$

Factor

$$-3\left(x-\frac{7-\sqrt{37}}{6}\right)\left(x-\frac{\sqrt{37}+7}{6}\right)$$

Steps Using the Quadratic Formula

Subtract 8 from 7 to get -1.

$$factor(-1+7x-3x^{2})$$

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.

$$-3x^{2}+7x-1=0$$

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.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-3\right)\left(-1\right)}}{2\left(-3\right)}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\left(-3\right)\left(-1\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-7±\sqrt{49+12\left(-1\right)}}{2\left(-3\right)}$$

Multiply 12 times -1.

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

Add 49 to -12.

$$x=\frac{-7±\sqrt{37}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-7±\sqrt{37}}{-6}$$

Now solve the equation x=\frac{-7±\sqrt{37}}{-6} when ± is plus. Add -7 to \sqrt{37}.

$$x=\frac{\sqrt{37}-7}{-6}$$

Divide -7+\sqrt{37} by -6.

$$x=\frac{7-\sqrt{37}}{6}$$

Now solve the equation x=\frac{-7±\sqrt{37}}{-6} when ± is minus. Subtract \sqrt{37} from -7.

$$x=\frac{-\sqrt{37}-7}{-6}$$

Divide -7-\sqrt{37} by -6.

$$x=\frac{\sqrt{37}+7}{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7-\sqrt{37}}{6} for x_{1} and \frac{7+\sqrt{37}}{6} for x_{2}.

$$-3x^{2}+7x-1=-3\left(x-\frac{7-\sqrt{37}}{6}\right)\left(x-\frac{\sqrt{37}+7}{6}\right)$$

Evaluate

$$-3x^{2}+7x-1$$

Solution Steps

Subtract 8 from 7 to get -1.

$$-1+7x-3x^{2}$$