Final answer:
To express 10b⁴ + 3b² - 11 in quadratic form, you can treat b² as a placeholder variable (t), which turns the expression into a quadratic equation 10t² + 3t - 11. Solve this using the quadratic formula for 't' and then revert 't' back to b² to find the original variable's values.
Step-by-step explanation:
To write 10b⁴ + 3b² - 11 in quadratic form, we need to recognize that this is not a standard quadratic equation because of the exponent on b's first term. Nonetheless, we can treat b² as a single variable (let's call it 't') for the sake of simplifying the expression. The given expression would then become:
10t² + 3t - 11,
which fits the standard quadratic form at² + bt + c. Here, 'a' is the coefficient of t², which is 10; 'b' is the coefficient of t, which is 3; and 'c' is the constant term, which is -11.
To solve for 't' using the quadratic formula, the formula is given by:
t = (-b ± √(b² - 4ac)) / (2a)
Applying the values from our quadratic form, we get:
t = (-3 ± √(3² - 4 × 10 × (-11))) / (2 × 10)
After calculating this, you would then replace 't' back with b² to find the values of 'b' for the original equation.