After being rearranged and simplified, which of the following equations could be solved using the quadratic formula? Check all that apply.
A. [tex]$2 x^2+x^2+x=30$[/tex]
B. [tex]$5 x+4=3 x^4-2$[/tex]
C. [tex]$-x^2+4 x+7=-x^2-9$[/tex]
D. [tex]$9 x+3 x^2=14+x-1$[/tex]
Answer (1)
Equation A simplifies to 3 x 2 + x − 30 = 0 , which is a quadratic equation.
Equation B simplifies to 3 x 4 − 5 x − 6 = 0 , which is not a quadratic equation.
Equation C simplifies to 4 x + 16 = 0 , which is a linear equation.
Equation D simplifies to 3 x 2 + 8 x − 13 = 0 , which is a quadratic equation.
The equations that can be solved using the quadratic formula are A and D. Therefore, the answer is A , D .
Explanation
Understanding the Quadratic Formula We need to determine which of the given equations can be rearranged into the standard quadratic form, which is a x 2 + b x + c = 0 , where a = 0 . If an equation can be rearranged into this form, it can be solved using the quadratic formula.
Analyzing Equation A Let's analyze each equation:
A. 2 x 2 + x 2 + x = 30 Combine like terms: 3 x 2 + x = 30 Rearrange to standard form: 3 x 2 + x − 30 = 0 This is a quadratic equation because it is in the form a x 2 + b x + c = 0 , where a = 3 , b = 1 , and c = − 30 .
Analyzing Equation B B. 5 x + 4 = 3 x 4 − 2 Rearrange the equation: 3 x 4 − 5 x − 6 = 0 This is not a quadratic equation because it has a term with x 4 . The highest power of x in a quadratic equation is 2.
Analyzing Equation C C. − x 2 + 4 x + 7 = − x 2 − 9 Add x 2 to both sides: 4 x + 7 = − 9 Subtract 7 from both sides: 4 x = − 16 Divide by 4: x = − 4 This is a linear equation, not a quadratic equation, because the highest power of x is 1.
Analyzing Equation D D. 9 x + 3 x 2 = 14 + x − 1 Simplify the right side: 9 x + 3 x 2 = 13 + x Rearrange to standard form: 3 x 2 + 9 x − x − 13 = 0 Combine like terms: 3 x 2 + 8 x − 13 = 0 This is a quadratic equation because it is in the form a x 2 + b x + c = 0 , where a = 3 , b = 8 , and c = − 13 .
Conclusion Therefore, equations A and D can be solved using the quadratic formula after being rearranged and simplified.
Examples
The quadratic formula is a powerful tool used in various fields, such as physics and engineering. For instance, when calculating the trajectory of a projectile, the quadratic formula can help determine the launch angle needed to hit a specific target. Similarly, in electrical engineering, it can be used to analyze circuits and determine the values of components needed to achieve a desired performance. Understanding quadratic equations and the quadratic formula is crucial for solving real-world problems in these fields.
