Solve (4-x)^2+(4-x)^2=(5/4)^2

$$( 4 - x ) ^ { 2 } + ( 4 - x ) ^ { 2 } = ( \frac { 5 } { 4 } ) ^ { 2 }$$

Solve for x

$$x=-\frac{5\sqrt{2}}{8}+4\approx 3.116116524$$

$$x=\frac{5\sqrt{2}}{8}+4\approx 4.883883476$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-x\right)^{2}.

$$16-8x+x^{2}+\left(4-x\right)^{2}=\left(\frac{5}{4}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-x\right)^{2}.

$$16-8x+x^{2}+16-8x+x^{2}=\left(\frac{5}{4}\right)^{2}$$

Add 16 and 16 to get 32.

$$32-8x+x^{2}-8x+x^{2}=\left(\frac{5}{4}\right)^{2}$$

Combine -8x and -8x to get -16x.

$$32-16x+x^{2}+x^{2}=\left(\frac{5}{4}\right)^{2}$$

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

$$32-16x+2x^{2}=\left(\frac{5}{4}\right)^{2}$$

Calculate \frac{5}{4} to the power of 2 and get \frac{25}{16}.

$$32-16x+2x^{2}=\frac{25}{16}$$

Subtract \frac{25}{16} from both sides.

$$32-16x+2x^{2}-\frac{25}{16}=0$$

Subtract \frac{25}{16} from 32 to get \frac{487}{16}.

$$\frac{487}{16}-16x+2x^{2}=0$$

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.

$$2x^{2}-16x+\frac{487}{16}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -16 for b, and \frac{487}{16} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 2\times \frac{487}{16}}}{2\times 2}$$

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\times 2\times \frac{487}{16}}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-16\right)±\sqrt{256-8\times \frac{487}{16}}}{2\times 2}$$

Multiply -8 times \frac{487}{16}.

$$x=\frac{-\left(-16\right)±\sqrt{256-\frac{487}{2}}}{2\times 2}$$

Add 256 to -\frac{487}{2}.

$$x=\frac{-\left(-16\right)±\sqrt{\frac{25}{2}}}{2\times 2}$$

Take the square root of \frac{25}{2}.

$$x=\frac{-\left(-16\right)±\frac{5\sqrt{2}}{2}}{2\times 2}$$

The opposite of -16 is 16.

$$x=\frac{16±\frac{5\sqrt{2}}{2}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{16±\frac{5\sqrt{2}}{2}}{4}$$

Now solve the equation x=\frac{16±\frac{5\sqrt{2}}{2}}{4} when ± is plus. Add 16 to \frac{5\sqrt{2}}{2}.

$$x=\frac{\frac{5\sqrt{2}}{2}+16}{4}$$

Divide 16+\frac{5\sqrt{2}}{2} by 4.

$$x=\frac{5\sqrt{2}}{8}+4$$

Now solve the equation x=\frac{16±\frac{5\sqrt{2}}{2}}{4} when ± is minus. Subtract \frac{5\sqrt{2}}{2} from 16.

$$x=\frac{-\frac{5\sqrt{2}}{2}+16}{4}$$

Divide 16-\frac{5\sqrt{2}}{2} by 4.

$$x=-\frac{5\sqrt{2}}{8}+4$$

The equation is now solved.

$$x=\frac{5\sqrt{2}}{8}+4 x=-\frac{5\sqrt{2}}{8}+4$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-x\right)^{2}.

$$16-8x+x^{2}+\left(4-x\right)^{2}=\left(\frac{5}{4}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-x\right)^{2}.

$$16-8x+x^{2}+16-8x+x^{2}=\left(\frac{5}{4}\right)^{2}$$

Add 16 and 16 to get 32.

$$32-8x+x^{2}-8x+x^{2}=\left(\frac{5}{4}\right)^{2}$$

Combine -8x and -8x to get -16x.

$$32-16x+x^{2}+x^{2}=\left(\frac{5}{4}\right)^{2}$$

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

$$32-16x+2x^{2}=\left(\frac{5}{4}\right)^{2}$$

Calculate \frac{5}{4} to the power of 2 and get \frac{25}{16}.

$$32-16x+2x^{2}=\frac{25}{16}$$

Subtract 32 from both sides.

$$-16x+2x^{2}=\frac{25}{16}-32$$

Subtract 32 from \frac{25}{16} to get -\frac{487}{16}.

$$-16x+2x^{2}=-\frac{487}{16}$$

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.

$$2x^{2}-16x=-\frac{487}{16}$$

Divide both sides by 2.

$$\frac{2x^{2}-16x}{2}=-\frac{\frac{487}{16}}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{16}{2}\right)x=-\frac{\frac{487}{16}}{2}$$

Divide -16 by 2.

$$x^{2}-8x=-\frac{\frac{487}{16}}{2}$$

Divide -\frac{487}{16} by 2.

$$x^{2}-8x=-\frac{487}{32}$$

Divide -8, the coefficient of the x term, by 2 to get -4. Then add the square of -4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-8x+\left(-4\right)^{2}=-\frac{487}{32}+\left(-4\right)^{2}$$

Square -4.

$$x^{2}-8x+16=-\frac{487}{32}+16$$

Add -\frac{487}{32} to 16.

$$x^{2}-8x+16=\frac{25}{32}$$

Factor x^{2}-8x+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-4\right)^{2}=\frac{25}{32}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-4\right)^{2}}=\sqrt{\frac{25}{32}}$$

Simplify.

$$x-4=\frac{5\sqrt{2}}{8} x-4=-\frac{5\sqrt{2}}{8}$$

Add 4 to both sides of the equation.

$$x=\frac{5\sqrt{2}}{8}+4 x=-\frac{5\sqrt{2}}{8}+4$$