Question

The steps to derive the quadratic formula are shown below:

Step 1 ax2 + bx + c = 0
Step 2 ax2 + bx = − c
Step 3 x2 + (b / a)x = - c / a
Step 4 x2 + (b / a)x + b ^2 / 4a^2 = (- c / a) + (b^2 / 4a^2)
Step 5 x2 + (b / a)x + b ^2 / 4a^2 = (-4ac / 4a^2) + (b^2/4a^2)
Step 6 x^2 + b / 2a = (b^2 - 4ac) / (4a^2)
Step 7 x + b / 2a = +- square root of b^2 - 4ac / 4a^2 = +- square root of b^2 - 4ac / 4a
Step 8 x = −b / 2a +- square root of b ^2 - 4ac / 2a
Step 9 x = (- b +- square root of b^2 - 4ac) / (2a)
Which of the following is the first incorrect step?
A) Step 4
B) Step 6
C) Step 7
D) Step 8

Answer 1

Step 3: B

Step 5: D

Step 6: A

Step 8: C

Explanation:

On edge

Answer 2

Step 6 is missing the fact that the left hand side is raised to the second power.

Explanation:

First cancel c from the left hand side by subtracting it from each side:

ax²+bx+c = 0

ax²+bx+c-c = 0-c

ax²+bx = -c

Next cancel a from the left hand side by dividing all terms by a:

ax²/a + bx/a = -c/a

x² + (b/a)x = -c/a

Next we will complete the square. We do this by dividing the second coefficient, b/a, by 2 and then squaring it; (b/a)÷2 = b/2a; (b/2a)² = b²/4a²

Add this to each side:

x²+(b/a)x+(b²/4a²) = -c/a + (b²/4a²)

Next we will find a common denominator on the right hand side. To do this, multiply the first term by 4a (to make the denominator 4a²):

x²+(b/a)x+(b²/4a²) = (-c*4a)/(a*4a) + (b²/4a²)

x²+(b/a)x+(b²/4a²) = -4ac/4a² + b²/4a²

We can write the left hand side as a squared binomial:

(x+b/2a)² = (b²-4ac)/4a²

Take the square root of both sides:

√(x+b/2a)² = √((b²-4ac)/4a²)

x+b/2a = √(b²-4ac)/2a

Subtract b/2a from each side:

x+b/2a - b/2a = √(b²-4ac)/2a - b/2a

x = (-b ± √(b²-4ac))/2a