Solve (x-1)^2-4(2-x)geq0

$$( x - 1 ) ^ { 2 } - 4 ( 2 - x ) \geq 0$$

Solve for x

$$x\in (-\infty,-2\sqrt{2}-1]\cup [2\sqrt{2}-1,\infty)$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$x^{2}-2x+1-4\left(2-x\right)\geq 0$$

Use the distributive property to multiply -4 by 2-x.

$$x^{2}-2x+1-8+4x\geq 0$$

Subtract 8 from 1 to get -7.

$$x^{2}-2x-7+4x\geq 0$$

Combine -2x and 4x to get 2x.

$$x^{2}+2x-7\geq 0$$

To solve the inequality, factor the left hand side. 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.

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

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, 2 for b, and -7 for c in the quadratic formula.

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

Do the calculations.

$$x=\frac{-2±4\sqrt{2}}{2}$$

Solve the equation x=\frac{-2±4\sqrt{2}}{2} when ± is plus and when ± is minus.

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

Rewrite the inequality by using the obtained solutions.

$$\left(x-\left(2\sqrt{2}-1\right)\right)\left(x-\left(-2\sqrt{2}-1\right)\right)\geq 0$$

For the product to be ≥0, x-\left(2\sqrt{2}-1\right) and x-\left(-2\sqrt{2}-1\right) have to be both ≤0 or both ≥0. Consider the case when x-\left(2\sqrt{2}-1\right) and x-\left(-2\sqrt{2}-1\right) are both ≤0.

$$x-\left(2\sqrt{2}-1\right)\leq 0 x-\left(-2\sqrt{2}-1\right)\leq 0$$

The solution satisfying both inequalities is x\leq -2\sqrt{2}-1.

$$x\leq -2\sqrt{2}-1$$

Consider the case when x-\left(2\sqrt{2}-1\right) and x-\left(-2\sqrt{2}-1\right) are both ≥0.

$$x-\left(-2\sqrt{2}-1\right)\geq 0 x-\left(2\sqrt{2}-1\right)\geq 0$$

The solution satisfying both inequalities is x\geq 2\sqrt{2}-1.

$$x\geq 2\sqrt{2}-1$$

The final solution is the union of the obtained solutions.

$$x\leq -2\sqrt{2}-1\text{; }x\geq 2\sqrt{2}-1$$