Solve 25x=sqrt{49x^2+4.8^2}

$$25x= \sqrt{ 49 { x }^{ 2 } + { 4.8 }^{ 2 } }$$

Solve for x

$$x=0.2$$

Solution Steps

Square both sides of the equation.

$$\left(25x\right)^{2}=\left(\sqrt{49x^{2}+4.8^{2}}\right)^{2}$$

Expand \left(25x\right)^{2}.

$$25^{2}x^{2}=\left(\sqrt{49x^{2}+4.8^{2}}\right)^{2}$$

Calculate 25 to the power of 2 and get 625.

$$625x^{2}=\left(\sqrt{49x^{2}+4.8^{2}}\right)^{2}$$

Calculate 4.8 to the power of 2 and get 23.04.

$$625x^{2}=\left(\sqrt{49x^{2}+23.04}\right)^{2}$$

Calculate \sqrt{49x^{2}+23.04} to the power of 2 and get 49x^{2}+23.04.

$$625x^{2}=49x^{2}+23.04$$

Subtract 49x^{2} from both sides.

$$625x^{2}-49x^{2}=23.04$$

Combine 625x^{2} and -49x^{2} to get 576x^{2}.

$$576x^{2}=23.04$$

Subtract 23.04 from both sides.

$$576x^{2}-23.04=0$$

Consider 576x^{2}-23.04. Rewrite 576x^{2}-23.04 as \left(24x\right)^{2}-\left(\frac{24}{5}\right)^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(24x-\frac{24}{5}\right)\left(24x+\frac{24}{5}\right)=0$$

To find equation solutions, solve 24x-\frac{24}{5}=0 and 24x+\frac{24}{5}=0.

$$x=\frac{1}{5} x=-\frac{1}{5}$$

Substitute \frac{1}{5} for x in the equation 25x=\sqrt{49x^{2}+4.8^{2}}.

$$25\times \frac{1}{5}=\sqrt{49\times \left(\frac{1}{5}\right)^{2}+4.8^{2}}$$

Simplify. The value x=\frac{1}{5} satisfies the equation.

$$5=5$$

Substitute -\frac{1}{5} for x in the equation 25x=\sqrt{49x^{2}+4.8^{2}}.

$$25\left(-\frac{1}{5}\right)=\sqrt{49\left(-\frac{1}{5}\right)^{2}+4.8^{2}}$$

Simplify. The value x=-\frac{1}{5} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-5=5$$

Equation 25x=\sqrt{49x^{2}+23.04} has a unique solution.

$$x=\frac{1}{5}$$