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

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

Evaluate

$$9-6x-4x^{2}$$

Solution Steps

Combine -7x and 4x to get -3x.

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

Combine -3x and -3x to get -6x.

$$x^{2}-6x+3+2-5x^{2}+4$$

Add 3 and 2 to get 5.

$$x^{2}-6x+5-5x^{2}+4$$

Combine x^{2} and -5x^{2} to get -4x^{2}.

$$-4x^{2}-6x+5+4$$

Add 5 and 4 to get 9.

$$-4x^{2}-6x+9$$

Factor

$$-4\left(x-\frac{-3\sqrt{5}-3}{4}\right)\left(x-\frac{3\sqrt{5}-3}{4}\right)$$

Steps Using the Quadratic Formula

Combine -7x and 4x to get -3x.

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

Combine -3x and -3x to get -6x.

$$factor(x^{2}-6x+3+2-5x^{2}+4)$$

Add 3 and 2 to get 5.

$$factor(x^{2}-6x+5-5x^{2}+4)$$

Combine x^{2} and -5x^{2} to get -4x^{2}.

$$factor(-4x^{2}-6x+5+4)$$

Add 5 and 4 to get 9.

$$factor(-4x^{2}-6x+9)$$

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.

$$-4x^{2}-6x+9=0$$

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.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-4\right)\times 9}}{2\left(-4\right)}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-4\right)\times 9}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-\left(-6\right)±\sqrt{36+16\times 9}}{2\left(-4\right)}$$

Multiply 16 times 9.

$$x=\frac{-\left(-6\right)±\sqrt{36+144}}{2\left(-4\right)}$$

Add 36 to 144.

$$x=\frac{-\left(-6\right)±\sqrt{180}}{2\left(-4\right)}$$

Take the square root of 180.

$$x=\frac{-\left(-6\right)±6\sqrt{5}}{2\left(-4\right)}$$

The opposite of -6 is 6.

$$x=\frac{6±6\sqrt{5}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{6±6\sqrt{5}}{-8}$$

Now solve the equation x=\frac{6±6\sqrt{5}}{-8} when ± is plus. Add 6 to 6\sqrt{5}.

$$x=\frac{6\sqrt{5}+6}{-8}$$

Divide 6+6\sqrt{5} by -8.

$$x=\frac{-3\sqrt{5}-3}{4}$$

Now solve the equation x=\frac{6±6\sqrt{5}}{-8} when ± is minus. Subtract 6\sqrt{5} from 6.

$$x=\frac{6-6\sqrt{5}}{-8}$$

Divide 6-6\sqrt{5} by -8.

$$x=\frac{3\sqrt{5}-3}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-3-3\sqrt{5}}{4} for x_{1} and \frac{-3+3\sqrt{5}}{4} for x_{2}.

$$-4x^{2}-6x+9=-4\left(x-\frac{-3\sqrt{5}-3}{4}\right)\left(x-\frac{3\sqrt{5}-3}{4}\right)$$