Solve 3x-2+-4x^2-5x+7

$$3 x - 2 + - 4 x ^ { 2 } - 5 x + 7$$

Evaluate

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

Solution Steps

Combine 3x and -5x to get -2x.

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

Add -2 and 7 to get 5.

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

Factor

$$-4\left(x-\frac{-\sqrt{21}-1}{4}\right)\left(x-\frac{\sqrt{21}-1}{4}\right)$$

Steps Using the Quadratic Formula

Combine 3x and -5x to get -2x.

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

Add -2 and 7 to get 5.

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

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.

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

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.

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

Square -2.

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

Multiply -4 times -4.

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

Multiply 16 times 5.

$$x=\frac{-\left(-2\right)±\sqrt{4+80}}{2\left(-4\right)}$$

Add 4 to 80.

$$x=\frac{-\left(-2\right)±\sqrt{84}}{2\left(-4\right)}$$

Take the square root of 84.

$$x=\frac{-\left(-2\right)±2\sqrt{21}}{2\left(-4\right)}$$

The opposite of -2 is 2.

$$x=\frac{2±2\sqrt{21}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{2±2\sqrt{21}}{-8}$$

Now solve the equation x=\frac{2±2\sqrt{21}}{-8} when ± is plus. Add 2 to 2\sqrt{21}.

$$x=\frac{2\sqrt{21}+2}{-8}$$

Divide 2+2\sqrt{21} by -8.

$$x=\frac{-\sqrt{21}-1}{4}$$

Now solve the equation x=\frac{2±2\sqrt{21}}{-8} when ± is minus. Subtract 2\sqrt{21} from 2.

$$x=\frac{2-2\sqrt{21}}{-8}$$

Divide 2-2\sqrt{21} by -8.

$$x=\frac{\sqrt{21}-1}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-1-\sqrt{21}}{4} for x_{1} and \frac{-1+\sqrt{21}}{4} for x_{2}.

$$-4x^{2}-2x+5=-4\left(x-\frac{-\sqrt{21}-1}{4}\right)\left(x-\frac{\sqrt{21}-1}{4}\right)$$