Question: Type the correct answer in each box. Solve the quadratic equation. Write your solutions in the blanks, starting with the smaller value. [tex]\[ 4x^2 - 7 = 9 \][/tex]
To solve the quadratic equation [tex]\( 4x^2 - 7 = 9 \)[/tex], follow these steps:
1. Start with the given equation: [tex]\[
4x^2 - 7 = 9
\][/tex]
2. Move all terms to one side to set the equation to zero: [tex]\[
4x^2 - 7 - 9 = 0
\][/tex] Simplify the left-hand side: [tex]\[
4x^2 - 16 = 0
\][/tex]
3. Factor the equation, if possible: Notice that [tex]\( 4x^2 - 16 \)[/tex] can be rewritten as: [tex]\[
4(x^2 - 4) = 0
\][/tex]
4. Set each factor to zero and solve for [tex]\( x \)[/tex]: First, divide by 4: [tex]\[
(x^2 - 4) = 0
\][/tex] Then solve the simpler quadratic equation: [tex]\[
x^2 - 4 = 0
\][/tex]
5. Factor [tex]\( x^2 - 4 \)[/tex] into its constituent parts: [tex]\[
(x - 2)(x + 2) = 0
\][/tex]
6. Solve each factor for [tex]\( x \)[/tex]: [tex]\[
x - 2 = 0 \quad \Rightarrow \quad x = 2
\][/tex] [tex]\[
x + 2 = 0 \quad \Rightarrow \quad x = -2
\][/tex]
7. List the solutions starting with the smaller value: [tex]\[
-2, 2
\][/tex]
So, the solutions to the quadratic equation [tex]\( 4x^2 - 7 = 9 \)[/tex] are [tex]\( \boxed{-2} \)[/tex] and [tex]\( \boxed{2} \)[/tex].
