Solve the equation by using the quadratic formula.
[tex]2 x^2-5 x=7[/tex]
A. [tex]x=\frac{2}{7},-1[/tex]
B. [tex]x=\frac{2}{7}, 0[/tex]
C. [tex]x=\frac{7}{2},-1[/tex]
D. [tex]x=\frac{2}{7}, 1[/tex]
Answer (2)
Rewrite the equation in standard form: 2 x 2 − 5 x − 7 = 0 .
Identify the coefficients: a = 2 , b = − 5 , c = − 7 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c = 4 5 ± 81 .
Calculate the roots: x = 2 7 , − 1 . The correct answer is x = 2 7 , − 1 .
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 2 x 2 − 5 x = 7 . Subtracting 7 from both sides, we get 2 x 2 − 5 x − 7 = 0 .
Identify coefficients Now, we identify the coefficients a , b , and c . In this case, a = 2 , b = − 5 , and c = − 7 .
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 ( 2 ) − ( − 5 ) ± ( − 5 ) 2 − 4 ( 2 ) ( − 7 )
x = 4 5 ± 25 + 56
x = 4 5 ± 81
x = 4 5 ± 9
Calculate the roots Now, we find the two possible values for x :
x 1 = 4 5 + 9 = 4 14 = 2 7
x 2 = 4 5 − 9 = 4 − 4 = − 1
Select the correct answer So, the solutions are x = 2 7 and x = − 1 . Comparing these solutions with the given options, we see that option c matches our solutions.
Examples
Quadratic equations are incredibly useful in various real-world applications. For instance, they are used in physics to model projectile motion, helping to determine the trajectory of a ball thrown in the air. In engineering, quadratic equations are used in designing bridges and arches, ensuring structural stability. Moreover, in finance, they can be used to model investment growth and calculate optimal investment strategies. Understanding quadratic equations provides a powerful tool for solving problems across many disciplines.
The solutions to the equation 2 x 2 − 5 x = 7 are x = 2 7 and x = − 1 . Therefore, the correct answer is option C. This was determined using the quadratic formula after rewriting the equation in standard form.
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