Solve (10-x)^2+6=x

$$( 10 - x ) ^ { 2 } + 6 = x$$

Solve for x

$$x = \frac{\sqrt{17} + 21}{2} \approx 12.561552813$$

$$x = \frac{21 - \sqrt{17}}{2} \approx 8.438447187$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(10-x\right)^{2}.

$$100-20x+x^{2}+6=x$$

Add 100 and 6 to get 106.

$$106-20x+x^{2}=x$$

Subtract x from both sides.

$$106-20x+x^{2}-x=0$$

Combine -20x and -x to get -21x.

$$106-21x+x^{2}=0$$

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.

$$x^{2}-21x+106=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -21 for b, and 106 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-21\right)±\sqrt{\left(-21\right)^{2}-4\times 106}}{2}$$

Square -21.

$$x=\frac{-\left(-21\right)±\sqrt{441-4\times 106}}{2}$$

Multiply -4 times 106.

$$x=\frac{-\left(-21\right)±\sqrt{441-424}}{2}$$

Add 441 to -424.

$$x=\frac{-\left(-21\right)±\sqrt{17}}{2}$$

The opposite of -21 is 21.

$$x=\frac{21±\sqrt{17}}{2}$$

Now solve the equation x=\frac{21±\sqrt{17}}{2} when ± is plus. Add 21 to \sqrt{17}.

$$x=\frac{\sqrt{17}+21}{2}$$

Now solve the equation x=\frac{21±\sqrt{17}}{2} when ± is minus. Subtract \sqrt{17} from 21.

$$x=\frac{21-\sqrt{17}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{17}+21}{2} x=\frac{21-\sqrt{17}}{2}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(10-x\right)^{2}.

$$100-20x+x^{2}+6=x$$

Add 100 and 6 to get 106.

$$106-20x+x^{2}=x$$

Subtract x from both sides.

$$106-20x+x^{2}-x=0$$

Combine -20x and -x to get -21x.

$$106-21x+x^{2}=0$$

Subtract 106 from both sides. Anything subtracted from zero gives its negation.

$$-21x+x^{2}=-106$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}-21x=-106$$

Divide -21, the coefficient of the x term, by 2 to get -\frac{21}{2}. Then add the square of -\frac{21}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-21x+\left(-\frac{21}{2}\right)^{2}=-106+\left(-\frac{21}{2}\right)^{2}$$

Square -\frac{21}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-21x+\frac{441}{4}=-106+\frac{441}{4}$$

Add -106 to \frac{441}{4}.

$$x^{2}-21x+\frac{441}{4}=\frac{17}{4}$$

Factor x^{2}-21x+\frac{441}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{21}{2}\right)^{2}=\frac{17}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{21}{2}\right)^{2}}=\sqrt{\frac{17}{4}}$$

Simplify.

$$x-\frac{21}{2}=\frac{\sqrt{17}}{2} x-\frac{21}{2}=-\frac{\sqrt{17}}{2}$$

Add \frac{21}{2} to both sides of the equation.

$$x=\frac{\sqrt{17}+21}{2} x=\frac{21-\sqrt{17}}{2}$$