Sure, let's solve the given equation step-by-step:
Given equation:
[tex]\[ -4x - 21 + x^2 = 0 \][/tex]
Step 1: Rewrite the equation in standard quadratic form:
[tex]\[ x^2 - 4x - 21 = 0 \][/tex]
Step 2: Factor the quadratic equation. We need to find two numbers that multiply to [tex]\(-21\)[/tex] and add up to [tex]\(-4\)[/tex]. These numbers are [tex]\( -7 \)[/tex] and [tex]\( 3 \)[/tex], because:
[tex]\[ (-7) \times (3) = -21 \][/tex]
[tex]\[ (-7) + (3) = -4 \][/tex]
So, we can factor the quadratic as:
[tex]\[ (x - 7)(x + 3) = 0 \][/tex]
Step 3: Set each factor equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ x - 7 = 0 \quad \Rightarrow \quad x = 7 \][/tex]
[tex]\[ x + 3 = 0 \quad \Rightarrow \quad x = -3 \][/tex]
Therefore, the solutions to the equation [tex]\( -4x - 21 + x^2 = 0 \)[/tex] are:
[tex]\[ x = -3 \][/tex]
[tex]\[ x = 7 \][/tex]
When placing the solutions in ascending order, we get:
[tex]\[ x = -3, 7 \][/tex]
So the solutions are:
[tex]\[ \boxed{-3} \quad \boxed{7} \][/tex]
