Solve 3x^2-4x^2-5x+8

$$3 x ^ { 2 } - 4 x ^ { 2 } - 5 x + 8$$

Factor

$$-\left(x-\frac{-\sqrt{57}-5}{2}\right)\left(x-\frac{\sqrt{57}-5}{2}\right)$$

Steps Using the Quadratic Formula

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

$$factor(-x^{2}-5x+8)$$

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.

$$-x^{2}-5x+8=0$$

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.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\left(-1\right)\times 8}}{2\left(-1\right)}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\left(-1\right)\times 8}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-5\right)±\sqrt{25+4\times 8}}{2\left(-1\right)}$$

Multiply 4 times 8.

$$x=\frac{-\left(-5\right)±\sqrt{25+32}}{2\left(-1\right)}$$

Add 25 to 32.

$$x=\frac{-\left(-5\right)±\sqrt{57}}{2\left(-1\right)}$$

The opposite of -5 is 5.

$$x=\frac{5±\sqrt{57}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{5±\sqrt{57}}{-2}$$

Now solve the equation x=\frac{5±\sqrt{57}}{-2} when ± is plus. Add 5 to \sqrt{57}.

$$x=\frac{\sqrt{57}+5}{-2}$$

Divide 5+\sqrt{57} by -2.

$$x=\frac{-\sqrt{57}-5}{2}$$

Now solve the equation x=\frac{5±\sqrt{57}}{-2} when ± is minus. Subtract \sqrt{57} from 5.

$$x=\frac{5-\sqrt{57}}{-2}$$

Divide 5-\sqrt{57} by -2.

$$x=\frac{\sqrt{57}-5}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-5-\sqrt{57}}{2} for x_{1} and \frac{-5+\sqrt{57}}{2} for x_{2}.

$$-x^{2}-5x+8=-\left(x-\frac{-\sqrt{57}-5}{2}\right)\left(x-\frac{\sqrt{57}-5}{2}\right)$$

Evaluate

$$8-5x-x^{2}$$

Solution Steps

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

$$-x^{2}-5x+8$$