Use the quadratic formula to solve the equation: [tex]$-3 x^2-x-3=0$[/tex]
Answer (2)
Identify coefficients: a = − 3 , b = − 1 , c = − 3 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c .
Substitute and simplify: x = 2 ( − 3 ) 1 ± ( − 1 ) 2 − 4 ( − 3 ) ( − 3 ) = − 6 1 ± − 35 .
Express the solution with imaginary unit: x = 6 − 1 ± i 35 .
Explanation
Understanding the Quadratic Formula We are given the quadratic equation − 3 x 2 − x − 3 = 0 and asked to solve it using the quadratic formula. The quadratic formula is a general method for finding the solutions (also called roots) of any quadratic equation of the form a x 2 + b x + c = 0 . The formula is given by:
x = 2 a − b ± b 2 − 4 a c
where a , b , and c are coefficients of the quadratic equation.
Identifying Coefficients In our equation, − 3 x 2 − x − 3 = 0 , we can identify the coefficients as follows:
a = − 3 b = − 1 c = − 3
Substituting into the Formula Now, we substitute these values into the quadratic formula:
x = 2 ( − 3 ) − ( − 1 ) ± ( − 1 ) 2 − 4 ( − 3 ) ( − 3 )
Simplifying the Expression Next, we simplify the expression step by step. First, simplify the terms inside the square root:
( − 1 ) 2 − 4 ( − 3 ) ( − 3 ) = 1 − 36 = − 35
So, we have:
x = − 6 1 ± − 35
Since the value inside the square root is negative, we will have complex solutions. We can rewrite − 35 as i 35 , where i is the imaginary unit ( i = − 1 ).
x = − 6 1 ± i 35
Final Simplification To make the expression look cleaner, we can multiply both the numerator and the denominator by -1:
x = 6 − 1 ∓ i 35
This is the same as:
x = 6 − 1 ± i 35
Final Answer Thus, the solutions to the quadratic equation − 3 x 2 − x − 3 = 0 are:
x = 6 − 1 + i 35 and x = 6 − 1 − i 35
So the final answer is:
x = 6 − 1 ± i 35
Examples
Quadratic equations are incredibly useful in physics, engineering, and even economics. For example, when designing a bridge, engineers use quadratic equations to calculate the curve of suspension cables. Similarly, in finance, quadratic equations can help model investment growth or calculate break-even points. Understanding how to solve these equations allows professionals to make informed decisions and create efficient designs.
To solve the equation − 3 x 2 − x − 3 = 0 , we use the quadratic formula and find the roots are complex numbers. The solutions are x = 6 − 1 + i 35 and x = 6 − 1 − i 35 . These roots indicate that the equation has no real solutions because the discriminant is negative.
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