Solve 4x^2+3x^2+2x-1

$$4 { x }^{ 2 } +3 { x }^{ 2 } +2x-1$$

Factor

$$7\left(x-\frac{-2\sqrt{2}-1}{7}\right)\left(x-\frac{2\sqrt{2}-1}{7}\right)$$

Steps Using the Quadratic Formula

Combine 4x^{2} and 3x^{2} to get 7x^{2}.

$$factor(7x^{2}+2x-1)$$

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.

$$7x^{2}+2x-1=0$$

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.

$$x=\frac{-2±\sqrt{2^{2}-4\times 7\left(-1\right)}}{2\times 7}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\times 7\left(-1\right)}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-2±\sqrt{4-28\left(-1\right)}}{2\times 7}$$

Multiply -28 times -1.

$$x=\frac{-2±\sqrt{4+28}}{2\times 7}$$

Add 4 to 28.

$$x=\frac{-2±\sqrt{32}}{2\times 7}$$

Take the square root of 32.

$$x=\frac{-2±4\sqrt{2}}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{-2±4\sqrt{2}}{14}$$

Now solve the equation x=\frac{-2±4\sqrt{2}}{14} when ± is plus. Add -2 to 4\sqrt{2}.

$$x=\frac{4\sqrt{2}-2}{14}$$

Divide 4\sqrt{2}-2 by 14.

$$x=\frac{2\sqrt{2}-1}{7}$$

Now solve the equation x=\frac{-2±4\sqrt{2}}{14} when ± is minus. Subtract 4\sqrt{2} from -2.

$$x=\frac{-4\sqrt{2}-2}{14}$$

Divide -2-4\sqrt{2} by 14.

$$x=\frac{-2\sqrt{2}-1}{7}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{2\sqrt{2}-1}{7} for x_{1} and \frac{-1-2\sqrt{2}}{7} for x_{2}.

$$7x^{2}+2x-1=7\left(x-\frac{2\sqrt{2}-1}{7}\right)\left(x-\frac{-2\sqrt{2}-1}{7}\right)$$

Evaluate

$$7x^{2}+2x-1$$

Solution Steps

Combine 4x^{2} and 3x^{2} to get 7x^{2}.

$$7x^{2}+2x-1$$