Solve 25+25x^2+9+left(21xright)^2=4+left(26xright)^2

$$25+25 { x }^{ 2 } +9+ { \left(21x \right) }^{ 2 } =4+ { \left(26x \right) }^{ 2 }$$

Solve for x

$$x=\frac{\sqrt{7}}{7}\approx 0.377964473$$

$$x=-\frac{\sqrt{7}}{7}\approx -0.377964473$$

Steps by Finding Square Root

Add 25 and 9 to get 34.

$$34+25x^{2}+\left(21x\right)^{2}=4+\left(26x\right)^{2}$$

Expand \left(21x\right)^{2}.

$$34+25x^{2}+21^{2}x^{2}=4+\left(26x\right)^{2}$$

Calculate 21 to the power of 2 and get 441.

$$34+25x^{2}+441x^{2}=4+\left(26x\right)^{2}$$

Combine 25x^{2} and 441x^{2} to get 466x^{2}.

$$34+466x^{2}=4+\left(26x\right)^{2}$$

Expand \left(26x\right)^{2}.

$$34+466x^{2}=4+26^{2}x^{2}$$

Calculate 26 to the power of 2 and get 676.

$$34+466x^{2}=4+676x^{2}$$

Subtract 676x^{2} from both sides.

$$34+466x^{2}-676x^{2}=4$$

Combine 466x^{2} and -676x^{2} to get -210x^{2}.

$$34-210x^{2}=4$$

Subtract 34 from both sides.

$$-210x^{2}=4-34$$

Subtract 34 from 4 to get -30.

$$-210x^{2}=-30$$

Divide both sides by -210.

$$x^{2}=\frac{-30}{-210}$$

Reduce the fraction \frac{-30}{-210} to lowest terms by extracting and canceling out -30.

$$x^{2}=\frac{1}{7}$$

Take the square root of both sides of the equation.

$$x=\frac{\sqrt{7}}{7} x=-\frac{\sqrt{7}}{7}$$

Steps Using the Quadratic Formula

Add 25 and 9 to get 34.

$$34+25x^{2}+\left(21x\right)^{2}=4+\left(26x\right)^{2}$$

Expand \left(21x\right)^{2}.

$$34+25x^{2}+21^{2}x^{2}=4+\left(26x\right)^{2}$$

Calculate 21 to the power of 2 and get 441.

$$34+25x^{2}+441x^{2}=4+\left(26x\right)^{2}$$

Combine 25x^{2} and 441x^{2} to get 466x^{2}.

$$34+466x^{2}=4+\left(26x\right)^{2}$$

Expand \left(26x\right)^{2}.

$$34+466x^{2}=4+26^{2}x^{2}$$

Calculate 26 to the power of 2 and get 676.

$$34+466x^{2}=4+676x^{2}$$

Subtract 4 from both sides.

$$34+466x^{2}-4=676x^{2}$$

Subtract 4 from 34 to get 30.

$$30+466x^{2}=676x^{2}$$

Subtract 676x^{2} from both sides.

$$30+466x^{2}-676x^{2}=0$$

Combine 466x^{2} and -676x^{2} to get -210x^{2}.

$$30-210x^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$-210x^{2}+30=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -210 for a, 0 for b, and 30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-210\right)\times 30}}{2\left(-210\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-210\right)\times 30}}{2\left(-210\right)}$$

Multiply -4 times -210.

$$x=\frac{0±\sqrt{840\times 30}}{2\left(-210\right)}$$

Multiply 840 times 30.

$$x=\frac{0±\sqrt{25200}}{2\left(-210\right)}$$

Take the square root of 25200.

$$x=\frac{0±60\sqrt{7}}{2\left(-210\right)}$$

Multiply 2 times -210.

$$x=\frac{0±60\sqrt{7}}{-420}$$

Now solve the equation x=\frac{0±60\sqrt{7}}{-420} when ± is plus.

$$x=-\frac{\sqrt{7}}{7}$$

Now solve the equation x=\frac{0±60\sqrt{7}}{-420} when ± is minus.

$$x=\frac{\sqrt{7}}{7}$$

The equation is now solved.

$$x=-\frac{\sqrt{7}}{7} x=\frac{\sqrt{7}}{7}$$