Solve left(5xright)^2=left(10-4xright)^2+4

$${ \left(5x \right) }^{ 2 } = { \left(10-4x \right) }^{ 2 } +4$$

Solve for x

$$x = \frac{2 \sqrt{634} - 40}{9} \approx 1.150968139$$

$$x=\frac{-2\sqrt{634}-40}{9}\approx -10.039857028$$

Steps Using the Quadratic Formula

Expand \left(5x\right)^{2}.

$$5^{2}x^{2}=\left(10-4x\right)^{2}+4$$

Calculate 5 to the power of 2 and get 25.

$$25x^{2}=\left(10-4x\right)^{2}+4$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(10-4x\right)^{2}.

$$25x^{2}=100-80x+16x^{2}+4$$

Add 100 and 4 to get 104.

$$25x^{2}=104-80x+16x^{2}$$

Subtract 104 from both sides.

$$25x^{2}-104=-80x+16x^{2}$$

Add 80x to both sides.

$$25x^{2}-104+80x=16x^{2}$$

Subtract 16x^{2} from both sides.

$$25x^{2}-104+80x-16x^{2}=0$$

Combine 25x^{2} and -16x^{2} to get 9x^{2}.

$$9x^{2}-104+80x=0$$

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.

$$9x^{2}+80x-104=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, 80 for b, and -104 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-80±\sqrt{80^{2}-4\times 9\left(-104\right)}}{2\times 9}$$

Square 80.

$$x=\frac{-80±\sqrt{6400-4\times 9\left(-104\right)}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-80±\sqrt{6400-36\left(-104\right)}}{2\times 9}$$

Multiply -36 times -104.

$$x=\frac{-80±\sqrt{6400+3744}}{2\times 9}$$

Add 6400 to 3744.

$$x=\frac{-80±\sqrt{10144}}{2\times 9}$$

Take the square root of 10144.

$$x=\frac{-80±4\sqrt{634}}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{-80±4\sqrt{634}}{18}$$

Now solve the equation x=\frac{-80±4\sqrt{634}}{18} when ± is plus. Add -80 to 4\sqrt{634}.

$$x=\frac{4\sqrt{634}-80}{18}$$

Divide -80+4\sqrt{634} by 18.

$$x=\frac{2\sqrt{634}-40}{9}$$

Now solve the equation x=\frac{-80±4\sqrt{634}}{18} when ± is minus. Subtract 4\sqrt{634} from -80.

$$x=\frac{-4\sqrt{634}-80}{18}$$

Divide -80-4\sqrt{634} by 18.

$$x=\frac{-2\sqrt{634}-40}{9}$$

The equation is now solved.

$$x=\frac{2\sqrt{634}-40}{9} x=\frac{-2\sqrt{634}-40}{9}$$

Steps for Completing the Square

Expand \left(5x\right)^{2}.

$$5^{2}x^{2}=\left(10-4x\right)^{2}+4$$

Calculate 5 to the power of 2 and get 25.

$$25x^{2}=\left(10-4x\right)^{2}+4$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(10-4x\right)^{2}.

$$25x^{2}=100-80x+16x^{2}+4$$

Add 100 and 4 to get 104.

$$25x^{2}=104-80x+16x^{2}$$

Add 80x to both sides.

$$25x^{2}+80x=104+16x^{2}$$

Subtract 16x^{2} from both sides.

$$25x^{2}+80x-16x^{2}=104$$

Combine 25x^{2} and -16x^{2} to get 9x^{2}.

$$9x^{2}+80x=104$$

Divide both sides by 9.

$$\frac{9x^{2}+80x}{9}=\frac{104}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$x^{2}+\frac{80}{9}x=\frac{104}{9}$$

Divide \frac{80}{9}, the coefficient of the x term, by 2 to get \frac{40}{9}. Then add the square of \frac{40}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{80}{9}x+\left(\frac{40}{9}\right)^{2}=\frac{104}{9}+\left(\frac{40}{9}\right)^{2}$$

Square \frac{40}{9} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{80}{9}x+\frac{1600}{81}=\frac{104}{9}+\frac{1600}{81}$$

Add \frac{104}{9} to \frac{1600}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{80}{9}x+\frac{1600}{81}=\frac{2536}{81}$$

Factor x^{2}+\frac{80}{9}x+\frac{1600}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{40}{9}\right)^{2}=\frac{2536}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{40}{9}\right)^{2}}=\sqrt{\frac{2536}{81}}$$

Simplify.

$$x+\frac{40}{9}=\frac{2\sqrt{634}}{9} x+\frac{40}{9}=-\frac{2\sqrt{634}}{9}$$

Subtract \frac{40}{9} from both sides of the equation.

$$x=\frac{2\sqrt{634}-40}{9} x=\frac{-2\sqrt{634}-40}{9}$$