To address the given problem, let's solve the equation [tex]\( x^2 - 6 = 16x + 30 \)[/tex] step by step:
### Step 1: Rewrite in Standard Form
Start with the original equation:
[tex]\[ x^2 - 6 = 16x + 30 \][/tex]
To rewrite this in standard form [tex]\( ax^2 + bx + c = 0 \)[/tex], first, move all terms to one side of the equation:
[tex]\[ x^2 - 6 - 16x - 30 = 0 \][/tex]
Combine like terms:
[tex]\[ x^2 - 16x - 36 = 0 \][/tex]
So, the equation in standard form is:
[tex]\[ x^2 - 16x - 36 = 0 \][/tex]
### Step 2: Factor the Quadratic Equation
Next, we need to factor the quadratic equation [tex]\( x^2 - 16x - 36 = 0 \)[/tex].
The factored form of the quadratic equation is:
[tex]\[ (x - 18)(x + 2) = 0 \][/tex]
### Step 3: Solve for [tex]\( x \)[/tex]
To find the values of [tex]\( x \)[/tex] that make the equation true, set each factor equal to zero and solve:
[tex]\[ x - 18 = 0 \][/tex]
[tex]\[ x = 18 \][/tex]
[tex]\[ x + 2 = 0 \][/tex]
[tex]\[ x = -2 \][/tex]
### Summary
The standard form of the equation is:
[tex]\[ x^2 - 16x - 36 = 0 \][/tex]
The factored form is:
[tex]\[ (x - 18)(x + 2) = 0 \][/tex]
The values of [tex]\( x \)[/tex] that make the equation true are:
[tex]\[ x = 18 \][/tex] and [tex]\[ x = -2 \][/tex]
