To solve the given quadratic equation, we will use algebraic manipulation and the method of solving quadratic equations. Here is a step-by-step solution:
Given equation:
[tex]\[ 4x^2 - 22 = x^2 + 5 \][/tex]
### Step 1: Move all terms to one side of the equation to set it equal to 0.
First, we need to move the terms from the right side of the equation to the left side. We do this by subtracting [tex]\(x^2\)[/tex] and 5 from both sides:
[tex]\[ 4x^2 - 22 - x^2 - 5 = 0 \][/tex]
### Step 2: Combine like terms.
Combine the [tex]\(x^2\)[/tex] terms and the constant terms:
[tex]\[ 4x^2 - x^2 - 22 - 5 = 0 \][/tex]
Simplify the equation:
[tex]\[ 3x^2 - 27 = 0 \][/tex]
### Step 3: Solve the quadratic equation [tex]\(3x^2 - 27 = 0\)[/tex].
First, add 27 to both sides to isolate the term involving [tex]\(x^2\)[/tex]:
[tex]\[ 3x^2 = 27 \][/tex]
Next, divide both sides by 3 to solve for [tex]\(x^2\)[/tex]:
[tex]\[ x^2 = 9 \][/tex]
### Step 4: Take the square root of both sides to solve for [tex]\(x\)[/tex].
[tex]\[ x = \pm\sqrt{9} \][/tex]
Since the square root of 9 is 3, we have:
[tex]\[ x = \pm 3 \][/tex]
Therefore, the solutions to the quadratic equation are:
[tex]\[ x = 3 \][/tex]
[tex]\[ x = -3 \][/tex]
### Conclusion:
The solutions to the quadratic equation [tex]\(4x^2 - 22 = x^2 + 5\)[/tex] are:
[tex]\[ x = 3 \][/tex]
[tex]\[ x = -3 \][/tex]
