Solve 6x^2-3x+5+4x-2x^2-6=

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

Evaluate

$$4x^{2}+x-1$$

Solution Steps

Combine -3x and 4x to get x.

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

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

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

Subtract 6 from 5 to get -1.

$$4x^{2}+x-1$$

Factor

$$4\left(x-\frac{-\sqrt{17}-1}{8}\right)\left(x-\frac{\sqrt{17}-1}{8}\right)$$

Steps Using the Quadratic Formula

Combine -3x and 4x to get x.

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

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

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

Subtract 6 from 5 to get -1.

$$factor(4x^{2}+x-1)$$

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.

$$4x^{2}+x-1=0$$

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.

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

Square 1.

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

Multiply -4 times 4.

$$x=\frac{-1±\sqrt{1-16\left(-1\right)}}{2\times 4}$$

Multiply -16 times -1.

$$x=\frac{-1±\sqrt{1+16}}{2\times 4}$$

Add 1 to 16.

$$x=\frac{-1±\sqrt{17}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-1±\sqrt{17}}{8}$$

Now solve the equation x=\frac{-1±\sqrt{17}}{8} when ± is plus. Add -1 to \sqrt{17}.

$$x=\frac{\sqrt{17}-1}{8}$$

Now solve the equation x=\frac{-1±\sqrt{17}}{8} when ± is minus. Subtract \sqrt{17} from -1.

$$x=\frac{-\sqrt{17}-1}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-1+\sqrt{17}}{8} for x_{1} and \frac{-1-\sqrt{17}}{8} for x_{2}.

$$4x^{2}+x-1=4\left(x-\frac{\sqrt{17}-1}{8}\right)\left(x-\frac{-\sqrt{17}-1}{8}\right)$$