April rewrote a quadratic function in vertex form.
h(x)=5^2-30x+30
step 1 h(x)=5(x^2-6x+ )+30
step 2 h(x)=5(x^2-6x+9)+30-45
step 3 h(x)= 5(x-3)^2+15
In which step did they make a mistake? and how? Please help
I see two errors
1. should be h(x)=5x^2-30x+30 and not h(x)=5^2-30x+30
2. in step 2 to step 3, +30-45 = -15 , not +15
you left the 1st x out of the original equation ... transcription mistake
but the error is in the last step ... the 15 should be negative
The mistake was made in step 2. Let's go through the steps again to identify the error:
Step 1: The initial given quadratic function is h(x) = 5x^2 - 30x + 30.
Step 2 (Mistake): The mistake is in completing the square. It seems that they forgot to divide -30 by 2 before squaring it. The correct step should be:
h(x) = 5(x^2 - 6x + 9) + 30.
After dividing -30 by 2, we get -15. When we square -15, we get 225, not 45. So, the "+30" term should not be changed to "-45."
Step 3: Once the mistake in step 2 is corrected, we have:
h(x) = 5(x - 3)^2 + 30.
Final Form: The correct vertex form of the quadratic function is h(x) = 5(x - 3)^2 + 30.
Therefore, the error occurred in step 2, where they incorrectly modified the "+30" term.
In the given steps, April is trying to rewrite the quadratic function in vertex form. Let's go through each step and identify any mistakes:
Step 1: h(x) = 5(x^2 - 6x + ) + 30
In this step, April correctly identified that the quadratic needs to be multiplied by 5 to make it a perfect square trinomial. However, she left an empty space for the constant term, which should complete the square. The constant term should be half the coefficient of the linear term, squared. Thus, it should be (6/2)^2 = 9.
Corrected step 1: h(x) = 5(x^2 - 6x + 9) + 30
Step 2: h(x) = 5(x^2 - 6x + 9) + 30 - 45
In this step, April is trying to combine like terms by subtracting 45 from 30. However, the correct value to subtract should be -45, not 45. This is because the previous expression is being subtracted, so it should be treated as a negative value.
Corrected step 2: h(x) = 5(x^2 - 6x + 9) - 15
Step 3: h(x) = 5(x - 3)^2 + 15
In this step, April successfully completed the square by factoring the quadratic expression as a perfect square trinomial. This correctly shows the quadratic function in vertex form.
So, the mistake was made in Step 2 when April subtracted 45 instead of -45 while combining like terms.