Solve x^2+47x^2+148x^2=1

$${ x }^{ 2 } +47 { x }^{ 2 } +148 { x }^{ 2 } = 1$$

Solve for x

$$x=\frac{1}{14}\approx 0.071428571$$

$$x=-\frac{1}{14}\approx -0.071428571$$

Steps Using Difference of Squares

Combine x^{2} and 47x^{2} to get 48x^{2}.

$$48x^{2}+148x^{2}=1$$

Combine 48x^{2} and 148x^{2} to get 196x^{2}.

$$196x^{2}=1$$

Subtract 1 from both sides.

$$196x^{2}-1=0$$

Consider 196x^{2}-1. Rewrite 196x^{2}-1 as \left(14x\right)^{2}-1^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(14x-1\right)\left(14x+1\right)=0$$

To find equation solutions, solve 14x-1=0 and 14x+1=0.

$$x=\frac{1}{14} x=-\frac{1}{14}$$

Steps by Finding Square Root

Combine x^{2} and 47x^{2} to get 48x^{2}.

$$48x^{2}+148x^{2}=1$$

Combine 48x^{2} and 148x^{2} to get 196x^{2}.

$$196x^{2}=1$$

Divide both sides by 196.

$$x^{2}=\frac{1}{196}$$

Take the square root of both sides of the equation.

$$x=\frac{1}{14} x=-\frac{1}{14}$$

Steps Using the Quadratic Formula

Combine x^{2} and 47x^{2} to get 48x^{2}.

$$48x^{2}+148x^{2}=1$$

Combine 48x^{2} and 148x^{2} to get 196x^{2}.

$$196x^{2}=1$$

Subtract 1 from both sides.

$$196x^{2}-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 196 for a, 0 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 196\left(-1\right)}}{2\times 196}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 196\left(-1\right)}}{2\times 196}$$

Multiply -4 times 196.

$$x=\frac{0±\sqrt{-784\left(-1\right)}}{2\times 196}$$

Multiply -784 times -1.

$$x=\frac{0±\sqrt{784}}{2\times 196}$$

Take the square root of 784.

$$x=\frac{0±28}{2\times 196}$$

Multiply 2 times 196.

$$x=\frac{0±28}{392}$$

Now solve the equation x=\frac{0±28}{392} when ± is plus. Reduce the fraction \frac{28}{392} to lowest terms by extracting and canceling out 28.

$$x=\frac{1}{14}$$

Now solve the equation x=\frac{0±28}{392} when ± is minus. Reduce the fraction \frac{-28}{392} to lowest terms by extracting and canceling out 28.

$$x=-\frac{1}{14}$$

The equation is now solved.

$$x=\frac{1}{14} x=-\frac{1}{14}$$