Solve 7-5x-3x^2+x-8+7x^2

$$7 - 5 x - 3 x ^ { 2 } + x - 8 + 7 x ^ { 2 }$$

Evaluate

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

Solution Steps

Combine -5x and x to get -4x.

$$7-4x-3x^{2}-8+7x^{2}$$

Subtract 8 from 7 to get -1.

$$-1-4x-3x^{2}+7x^{2}$$

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

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

Factor

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

Steps Using the Quadratic Formula

Combine -5x and x to get -4x.

$$factor(7-4x-3x^{2}-8+7x^{2})$$

Subtract 8 from 7 to get -1.

$$factor(-1-4x-3x^{2}+7x^{2})$$

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

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

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.

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

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.

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

Square -4.

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

Multiply -4 times 4.

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

Multiply -16 times -1.

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

Add 16 to 16.

$$x=\frac{-\left(-4\right)±\sqrt{32}}{2\times 4}$$

Take the square root of 32.

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

The opposite of -4 is 4.

$$x=\frac{4±4\sqrt{2}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{4±4\sqrt{2}}{8}$$

Now solve the equation x=\frac{4±4\sqrt{2}}{8} when ± is plus. Add 4 to 4\sqrt{2}.

$$x=\frac{4\sqrt{2}+4}{8}$$

Divide 4+4\sqrt{2} by 8.

$$x=\frac{\sqrt{2}+1}{2}$$

Now solve the equation x=\frac{4±4\sqrt{2}}{8} when ± is minus. Subtract 4\sqrt{2} from 4.

$$x=\frac{4-4\sqrt{2}}{8}$$

Divide 4-4\sqrt{2} by 8.

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

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

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