Solve left(6xright)^2+left(6-5xright)^2=36

$${ \left(6x \right) }^{ 2 } + { \left(6-5x \right) }^{ 2 } =36$$

Solve for x

$$x=\frac{60}{61}\approx 0.983606557$$

$$x=0$$

Steps Using Factoring

Expand \left(6x\right)^{2}.

$$6^{2}x^{2}+\left(6-5x\right)^{2}=36$$

Calculate 6 to the power of 2 and get 36.

$$36x^{2}+\left(6-5x\right)^{2}=36$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(6-5x\right)^{2}.

$$36x^{2}+36-60x+25x^{2}=36$$

Combine 36x^{2} and 25x^{2} to get 61x^{2}.

$$61x^{2}+36-60x=36$$

Subtract 36 from both sides.

$$61x^{2}+36-60x-36=0$$

Subtract 36 from 36 to get 0.

$$61x^{2}-60x=0$$

Factor out x.

$$x\left(61x-60\right)=0$$

To find equation solutions, solve x=0 and 61x-60=0.

$$x=0 x=\frac{60}{61}$$

Steps Using the Quadratic Formula

Expand \left(6x\right)^{2}.

$$6^{2}x^{2}+\left(6-5x\right)^{2}=36$$

Calculate 6 to the power of 2 and get 36.

$$36x^{2}+\left(6-5x\right)^{2}=36$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(6-5x\right)^{2}.

$$36x^{2}+36-60x+25x^{2}=36$$

Combine 36x^{2} and 25x^{2} to get 61x^{2}.

$$61x^{2}+36-60x=36$$

Subtract 36 from both sides.

$$61x^{2}+36-60x-36=0$$

Subtract 36 from 36 to get 0.

$$61x^{2}-60x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 61 for a, -60 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-60\right)±\sqrt{\left(-60\right)^{2}}}{2\times 61}$$

Take the square root of \left(-60\right)^{2}.

$$x=\frac{-\left(-60\right)±60}{2\times 61}$$

The opposite of -60 is 60.

$$x=\frac{60±60}{2\times 61}$$

Multiply 2 times 61.

$$x=\frac{60±60}{122}$$

Now solve the equation x=\frac{60±60}{122} when ± is plus. Add 60 to 60.

$$x=\frac{120}{122}$$

Reduce the fraction \frac{120}{122} to lowest terms by extracting and canceling out 2.

$$x=\frac{60}{61}$$

Now solve the equation x=\frac{60±60}{122} when ± is minus. Subtract 60 from 60.

$$x=\frac{0}{122}$$

Divide 0 by 122.

$$x=0$$

The equation is now solved.

$$x=\frac{60}{61} x=0$$

Steps for Completing the Square

Expand \left(6x\right)^{2}.

$$6^{2}x^{2}+\left(6-5x\right)^{2}=36$$

Calculate 6 to the power of 2 and get 36.

$$36x^{2}+\left(6-5x\right)^{2}=36$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(6-5x\right)^{2}.

$$36x^{2}+36-60x+25x^{2}=36$$

Combine 36x^{2} and 25x^{2} to get 61x^{2}.

$$61x^{2}+36-60x=36$$

Subtract 36 from both sides.

$$61x^{2}-60x=36-36$$

Subtract 36 from 36 to get 0.

$$61x^{2}-60x=0$$

Divide both sides by 61.

$$\frac{61x^{2}-60x}{61}=\frac{0}{61}$$

Dividing by 61 undoes the multiplication by 61.

$$x^{2}-\frac{60}{61}x=\frac{0}{61}$$

Divide 0 by 61.

$$x^{2}-\frac{60}{61}x=0$$

Divide -\frac{60}{61}, the coefficient of the x term, by 2 to get -\frac{30}{61}. Then add the square of -\frac{30}{61} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{60}{61}x+\left(-\frac{30}{61}\right)^{2}=\left(-\frac{30}{61}\right)^{2}$$

Square -\frac{30}{61} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{60}{61}x+\frac{900}{3721}=\frac{900}{3721}$$

Factor x^{2}-\frac{60}{61}x+\frac{900}{3721}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{30}{61}\right)^{2}=\frac{900}{3721}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{30}{61}\right)^{2}}=\sqrt{\frac{900}{3721}}$$

Simplify.

$$x-\frac{30}{61}=\frac{30}{61} x-\frac{30}{61}=-\frac{30}{61}$$

Add \frac{30}{61} to both sides of the equation.

$$x=\frac{60}{61} x=0$$