underoot 2x square +3x + underoot 2 by alpha into beta method.
To solve the quadratic equation given as √(2x^2 + 3x + √2), we can use the method known as the alpha and beta method, also known as factoring or splitting the middle term.
Step 1: Write the equation in the standard quadratic form: ax^2 + bx + c = 0.
In this case, the equation is √(2x^2 + 3x + √2).
Let's assume the equation is equal to zero to simplify it.
√(2x^2 + 3x + √2) = 0
Step 2: Square both sides of the equation to eliminate the square root sign.
(√(2x^2 + 3x + √2))^2 = 0^2
2x^2 + 3x + √2 = 0
Step 3: Rearrange the equation so that it is in standard quadratic form.
2x^2 + 3x + √2 = 0
Step 4: To factor the quadratic equation, we need to find two numbers (say, α and β) such that:
- Their sum is equal to the coefficient of the middle term (3 in our case): α + β = 3
- Their product is equal to the product of the coefficient of the squared term and the constant term (2 times √2): α * β = 2√2
Step 5: By inspection, we can see that the numbers 1 and 2 satisfy the above conditions: 1 + 2 = 3 and 1 * 2 = 2.
Therefore, α = 1 and β = 2.
Step 6: Now, rewrite the middle term (3x) using the values of α and β obtained in the previous step.
2x^2 + (α + β)x + √2 = 0
2x^2 + (1 + 2)x + √2 = 0
Simplifying further:
2x^2 + 3x + √2 = 0
Step 7: Now, we can split the middle term using the values of α and β:
2x^2 + αx + βx + √2 = 0
Group the terms:
(2x^2 + αx) + (βx + √2) = 0
Step 8: Factor out the common terms from each group:
x(2x + α) + (√2 + β) = 0
Step 9: Finally, we obtain the factored form of the quadratic equation:
x(2x + α) + (√2 + β) = 0
Step 10: Set each factor equal to zero and solve for x:
x = 0 (from x = 0)
or
2x + α = 0 (from 2x + α = 0)
x = -α/2
Thus, the solution to the quadratic equation √(2x^2 + 3x + √2) = 0 using the alpha and beta method is:
x = 0 or x = -α/2