Solve x^2+left(5-xright)^2-16=x(5-x)

$${ x }^{ 2 } + { \left(5-x \right) }^{ 2 } -16=x(5-x)$$

Solve for x

$$x = \frac{\sqrt{13} + 5}{2} \approx 4.302775638$$

$$x=\frac{5-\sqrt{13}}{2}\approx 0.697224362$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5-x\right)^{2}.

$$x^{2}+25-10x+x^{2}-16=x\left(5-x\right)$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}+25-10x-16=x\left(5-x\right)$$

Subtract 16 from 25 to get 9.

$$2x^{2}+9-10x=x\left(5-x\right)$$

Use the distributive property to multiply x by 5-x.

$$2x^{2}+9-10x=5x-x^{2}$$

Subtract 5x from both sides.

$$2x^{2}+9-10x-5x=-x^{2}$$

Combine -10x and -5x to get -15x.

$$2x^{2}+9-15x=-x^{2}$$

Add x^{2} to both sides.

$$2x^{2}+9-15x+x^{2}=0$$

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

$$3x^{2}+9-15x=0$$

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.

$$3x^{2}-15x+9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -15 for b, and 9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times 3\times 9}}{2\times 3}$$

Square -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\times 3\times 9}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-15\right)±\sqrt{225-12\times 9}}{2\times 3}$$

Multiply -12 times 9.

$$x=\frac{-\left(-15\right)±\sqrt{225-108}}{2\times 3}$$

Add 225 to -108.

$$x=\frac{-\left(-15\right)±\sqrt{117}}{2\times 3}$$

Take the square root of 117.

$$x=\frac{-\left(-15\right)±3\sqrt{13}}{2\times 3}$$

The opposite of -15 is 15.

$$x=\frac{15±3\sqrt{13}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{15±3\sqrt{13}}{6}$$

Now solve the equation x=\frac{15±3\sqrt{13}}{6} when ± is plus. Add 15 to 3\sqrt{13}.

$$x=\frac{3\sqrt{13}+15}{6}$$

Divide 15+3\sqrt{13} by 6.

$$x=\frac{\sqrt{13}+5}{2}$$

Now solve the equation x=\frac{15±3\sqrt{13}}{6} when ± is minus. Subtract 3\sqrt{13} from 15.

$$x=\frac{15-3\sqrt{13}}{6}$$

Divide 15-3\sqrt{13} by 6.

$$x=\frac{5-\sqrt{13}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{13}+5}{2} x=\frac{5-\sqrt{13}}{2}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5-x\right)^{2}.

$$x^{2}+25-10x+x^{2}-16=x\left(5-x\right)$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}+25-10x-16=x\left(5-x\right)$$

Subtract 16 from 25 to get 9.

$$2x^{2}+9-10x=x\left(5-x\right)$$

Use the distributive property to multiply x by 5-x.

$$2x^{2}+9-10x=5x-x^{2}$$

Subtract 5x from both sides.

$$2x^{2}+9-10x-5x=-x^{2}$$

Combine -10x and -5x to get -15x.

$$2x^{2}+9-15x=-x^{2}$$

Add x^{2} to both sides.

$$2x^{2}+9-15x+x^{2}=0$$

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

$$3x^{2}+9-15x=0$$

Subtract 9 from both sides. Anything subtracted from zero gives its negation.

$$3x^{2}-15x=-9$$

Divide both sides by 3.

$$\frac{3x^{2}-15x}{3}=-\frac{9}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\left(-\frac{15}{3}\right)x=-\frac{9}{3}$$

Divide -15 by 3.

$$x^{2}-5x=-\frac{9}{3}$$

Divide -9 by 3.

$$x^{2}-5x=-3$$

Divide -5, the coefficient of the x term, by 2 to get -\frac{5}{2}. Then add the square of -\frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=-3+\left(-\frac{5}{2}\right)^{2}$$

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-5x+\frac{25}{4}=-3+\frac{25}{4}$$

Add -3 to \frac{25}{4}.

$$x^{2}-5x+\frac{25}{4}=\frac{13}{4}$$

Factor x^{2}-5x+\frac{25}{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}.

$$\left(x-\frac{5}{2}\right)^{2}=\frac{13}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{\frac{13}{4}}$$

Simplify.

$$x-\frac{5}{2}=\frac{\sqrt{13}}{2} x-\frac{5}{2}=-\frac{\sqrt{13}}{2}$$

Add \frac{5}{2} to both sides of the equation.

$$x=\frac{\sqrt{13}+5}{2} x=\frac{5-\sqrt{13}}{2}$$