solve the following quadratic equation for all values of x in simplest form. 16(x² - 4) - 5 = 12

Explanation:

Step1: Simplify the equation

First, we simplify the left - hand side of the equation (16(x^{2}-4)-5 = 12). Expand (16(x^{2}-4)) using the distributive property (a(b - c)=ab - ac), where (a = 16), (b=x^{2}) and (c = 4). So (16(x^{2}-4)=16x^{2}-64). The equation becomes (16x^{2}-64 - 5=12). Combine like terms: (-64-5=-69), so the equation is (16x^{2}-69 = 12).

Step2: Isolate the (x^{2}) term

Add 69 to both sides of the equation (16x^{2}-69 = 12). We get (16x^{2}=12 + 69). Calculate the right - hand side: (12+69 = 81), so (16x^{2}=81).

Step3: Solve for (x^{2})

Divide both sides of the equation (16x^{2}=81) by 16. We have (x^{2}=\frac{81}{16}).

Step4: Solve for (x)

Take the square root of both sides. Remember that if (x^{2}=a) ((a\geq0)), then (x=\pm\sqrt{a}). So (x=\pm\sqrt{\frac{81}{16}}). Since (\sqrt{\frac{81}{16}}=\frac{\sqrt{81}}{\sqrt{16}}=\frac{9}{4}), we get (x=\pm\frac{9}{4}).

Answer:

(x = \pm\frac{9}{4})