Solve for x.
3x = 6x - 2
Answer (2)
Expand the given equation y = ( 2 x + 3 ) 2 to get y = 4 x 2 + 12 x + 9 .
Rewrite the equation in standard form as 0 = 4 x 2 + 12 x + 9 .
Identify the coefficients a , b , and c from the standard form.
The values are a = 4 , b = 12 , and c = 9 , so the final answer is a = 4 , b = 12 , c = 9 .
Explanation
Understanding the Problem We are given the quadratic equation y = ( 2 x + 3 ) 2 . Our goal is to express this equation in standard form and then identify the values of a , b , and c in the standard form 0 = a x 2 + b x + c .
Expanding the Equation To find the standard form, we need to expand the given equation. We have:
y = ( 2 x + 3 ) 2 = ( 2 x + 3 ) ( 2 x + 3 )
Expanding the Product Now, let's expand the product:
y = ( 2 x ) ( 2 x ) + ( 2 x ) ( 3 ) + ( 3 ) ( 2 x ) + ( 3 ) ( 3 ) = 4 x 2 + 6 x + 6 x + 9 = 4 x 2 + 12 x + 9
Rewriting in Standard Form So, we have y = 4 x 2 + 12 x + 9 . To express this in the standard form 0 = a x 2 + b x + c , we can rewrite it as:
0 = 4 x 2 + 12 x + 9 − y
Since we want the equation to be equal to zero, we set y = 0 :
0 = 4 x 2 + 12 x + 9
Choosing the Correct Standard Form Comparing this with the given options, we see that the correct standard form is 0 = 4 x 2 + 12 x + 9 .
Identifying the Coefficients Now, we identify the coefficients a , b , and c from the standard form 0 = a x 2 + b x + c . In this case, we have:
a = 4 b = 12 c = 9
Final Answer Therefore, the standard form of the given quadratic equation is 0 = 4 x 2 + 12 x + 9 , and the values of a , b , and c are a = 4 , b = 12 , and c = 9 .
Examples
Understanding quadratic equations is crucial in various fields, such as physics and engineering. For instance, when designing a bridge, engineers use quadratic equations to model the parabolic shape of the bridge's arch. The coefficients a , b , and c in the quadratic equation help determine the bridge's height, width, and overall stability. By accurately calculating these values, engineers can ensure the bridge's structural integrity and safety. Similarly, in physics, quadratic equations are used to describe projectile motion, where the height of a projectile is modeled as a quadratic function of time.
To solve the equation 3 x = 6 x − 2 , we isolate x by rearranging and simplifying, leading to x = 3 2 . This solution can be verified by substituting it back into the original equation.
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