Solve f(x)=2x^2-12/2x^2-5x+3

$$f ( x ) = 2 x ^ { 2 } - \frac { 12 } { 2 } x ^ { 2 } - 5 x + 3$$

Evaluate

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

Solution Steps

Divide 12 by 2 to get 6.

$$2x^{2}-6x^{2}-5x+3$$

Combine 2x^{2} and -6x^{2} to get -4x^{2}.

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

Factor

$$-4\left(x-\frac{-\sqrt{73}-5}{8}\right)\left(x-\frac{\sqrt{73}-5}{8}\right)$$

Steps Using the Quadratic Formula

Divide 12 by 2 to get 6.

$$factor(2x^{2}-6x^{2}-5x+3)$$

Combine 2x^{2} and -6x^{2} to get -4x^{2}.

$$factor(-4x^{2}-5x+3)$$

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.

$$-4x^{2}-5x+3=0$$

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.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\left(-4\right)\times 3}}{2\left(-4\right)}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\left(-4\right)\times 3}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-\left(-5\right)±\sqrt{25+16\times 3}}{2\left(-4\right)}$$

Multiply 16 times 3.

$$x=\frac{-\left(-5\right)±\sqrt{25+48}}{2\left(-4\right)}$$

Add 25 to 48.

$$x=\frac{-\left(-5\right)±\sqrt{73}}{2\left(-4\right)}$$

The opposite of -5 is 5.

$$x=\frac{5±\sqrt{73}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{5±\sqrt{73}}{-8}$$

Now solve the equation x=\frac{5±\sqrt{73}}{-8} when ± is plus. Add 5 to \sqrt{73}.

$$x=\frac{\sqrt{73}+5}{-8}$$

Divide 5+\sqrt{73} by -8.

$$x=\frac{-\sqrt{73}-5}{8}$$

Now solve the equation x=\frac{5±\sqrt{73}}{-8} when ± is minus. Subtract \sqrt{73} from 5.

$$x=\frac{5-\sqrt{73}}{-8}$$

Divide 5-\sqrt{73} by -8.

$$x=\frac{\sqrt{73}-5}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-5-\sqrt{73}}{8} for x_{1} and \frac{-5+\sqrt{73}}{8} for x_{2}.

$$-4x^{2}-5x+3=-4\left(x-\frac{-\sqrt{73}-5}{8}\right)\left(x-\frac{\sqrt{73}-5}{8}\right)$$