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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]

Solution: [tex]\(\square\)[/tex] , [tex]\(\square\)[/tex]


Sagot :

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].