Solve the equation by using the quadratic formula:
[tex]3 x^2-1=7 x[/tex]
a. [tex]\frac{7+\sqrt{37}}{6}, \frac{7-\sqrt{37}}{6}[/tex]
c. [tex]\frac{-7+\sqrt{37}}{6}, \frac{-7-\sqrt{37}}{6}[/tex]
b. [tex]\frac{7+\sqrt{61}}{6}, \frac{7-\sqrt{61}}{6}[/tex]
d. [tex]\frac{-7+\sqrt{61}}{-6}, \frac{-7-\sqrt{61}}{6}[/tex]
Please select the best answer from the choices provided.
Answer (2)
The solutions for the equation are x = 6 7 + 61 and x = 6 7 − 61 , which corresponds to option b. We used the quadratic formula after converting the equation into standard form. The coefficients were identified, and calculations followed to find the roots.
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Rewrite the equation in standard form: 3 x 2 − 7 x − 1 = 0 .
Identify the coefficients: a = 3 , b = − 7 , c = − 1 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c = 6 7 ± 61 .
The solutions are x = 6 7 + 61 and x = 6 7 − 61 , so the answer is b .
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 3 x 2 − 1 = 7 x . Subtracting 7 x from both sides, we get 3 x 2 − 7 x − 1 = 0 .
Identify coefficients Now, we identify the coefficients a , b , and c . In the equation 3 x 2 − 7 x − 1 = 0 , we have a = 3 , b = − 7 , and c = − 1 .
Apply quadratic formula Next, we apply the quadratic formula, which is given by x = 2 a − b ± b 2 − 4 a c . Substituting the values of a , b , and c , we get
x = 2 ( 3 ) − ( − 7 ) ± ( − 7 ) 2 − 4 ( 3 ) ( − 1 )
x = 6 7 ± 49 + 12
x = 6 7 ± 61
Find the solutions So the two solutions are x = 6 7 + 61 and x = 6 7 − 61 . Comparing these with the given options, we see that option b matches our solutions.
Examples
Quadratic equations are used in various real-life scenarios, such as calculating the trajectory of a projectile, determining the dimensions of a rectangular area given its area and a relationship between its sides, or modeling the growth of a population. For instance, if you want to build a rectangular garden with an area of 100 square meters and the length must be 5 meters more than the width, you can use a quadratic equation to find the dimensions of the garden.
