Solve 4x^2-169=0

$$4 x ^ { 2 } - 169 = 0$$

Solve for x

$$x = \frac{13}{2} = 6\frac{1}{2} = 6.5$$

$$x = -\frac{13}{2} = -6\frac{1}{2} = -6.5$$

Steps Using Difference of Squares

Consider 4x^{2}-169. Rewrite 4x^{2}-169 as \left(2x\right)^{2}-13^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(2x-13\right)\left(2x+13\right)=0$$

To find equation solutions, solve 2x-13=0 and 2x+13=0.

$$x=\frac{13}{2} x=-\frac{13}{2}$$

Steps by Finding Square Root

Add 169 to both sides. Anything plus zero gives itself.

$$4x^{2}=169$$

Divide both sides by 4.

$$x^{2}=\frac{169}{4}$$

Take the square root of both sides of the equation.

$$x=\frac{13}{2} x=-\frac{13}{2}$$

Steps Using the Quadratic Formula

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.

$$4x^{2}-169=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 0 for b, and -169 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 4\left(-169\right)}}{2\times 4}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 4\left(-169\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{0±\sqrt{-16\left(-169\right)}}{2\times 4}$$

Multiply -16 times -169.

$$x=\frac{0±\sqrt{2704}}{2\times 4}$$

Take the square root of 2704.

$$x=\frac{0±52}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{0±52}{8}$$

Now solve the equation x=\frac{0±52}{8} when ± is plus. Reduce the fraction \frac{52}{8} to lowest terms by extracting and canceling out 4.

$$x=\frac{13}{2}$$

Now solve the equation x=\frac{0±52}{8} when ± is minus. Reduce the fraction \frac{-52}{8} to lowest terms by extracting and canceling out 4.

$$x=-\frac{13}{2}$$

The equation is now solved.

$$x=\frac{13}{2} x=-\frac{13}{2}$$