Use the quadratic formula to solve for [tex]$x$[/tex].
[tex]$6 x^2+7 x=2$[/tex]
Answer (1)
Rewrite the equation in standard form: 6 x 2 + 7 x − 2 = 0 .
Identify the coefficients: a = 6 , b = 7 , c = − 2 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c = 12 − 7 ± 97 .
The solutions are: x = 12 − 7 − 97 and x = 12 − 7 + 97 , which are approximately x ≈ − 1.404 and x ≈ 0.237 .
x = 12 − 7 ± 97
Explanation
Rewrite the equation First, we need to rewrite the given equation in the standard quadratic form, which is a x 2 + b x + c = 0 . The given equation is 6 x 2 + 7 x = 2 . Subtracting 2 from both sides, we get 6 x 2 + 7 x − 2 = 0 .
Identify coefficients Now, we can identify the coefficients a , b , and c . In this case, a = 6 , b = 7 , and c = − 2 .
Apply quadratic formula Next, we substitute these values into the quadratic formula, which is given by x = 2 a − b ± b 2 − 4 a c . Substituting a = 6 , b = 7 , and c = − 2 , we get:
x = 2 ( 6 ) − 7 ± 7 2 − 4 ( 6 ) ( − 2 )
x = 12 − 7 ± 49 + 48
x = 12 − 7 ± 97
Find the solutions So the two solutions for x are:
x 1 = 12 − 7 − 97
x 2 = 12 − 7 + 97
We can approximate the values of x :
x 1 ≈ 12 − 7 − 9.85 ≈ − 1.404
x 2 ≈ 12 − 7 + 9.85 ≈ 0.237
Final Answer Therefore, the solutions for x are approximately x ≈ − 1.404 and x ≈ 0.237 .
Examples
The quadratic formula is not just an abstract concept; it has real-world applications. For instance, engineers use it to calculate the trajectory of a projectile, economists use it to model supply and demand curves, and financial analysts use it to optimize investment portfolios. Imagine designing a bridge where the curve of the arch is described by a quadratic equation; the quadratic formula helps determine the precise points where the arch meets the supports, ensuring structural integrity. Understanding and applying the quadratic formula allows professionals to solve complex problems and make informed decisions in various fields.
