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Given an equation $6 + 4x - 2x^2 = 0$ . (a) Express $6 + 4x - 2x^2$ in the form $a - (x + b)^2$

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Find equations of tangent lines using limits

Full solution

Q. Given an equation

6 + 4x - 2x^2 = 0

. (a) Express

6 + 4x - 2x^2

in the form

a - (x + b)^2

Arrange in Descending Order: We need to complete the square to express the given quadratic equation in the form $a - (x + b)^2$ . The first step is to arrange the equation in descending order of the powers of $x$ . $\newline$ $6 + 4x - 2x^2$ rearranged is $-2x^2 + 4x + 6$ .
Factor Out Coefficient: Next, we factor out the coefficient of the $x^2$ term from the $x^2$ and $x$ terms. $\newline$ This gives us $-2(x^2 - 2x) + 6$ .
Find Value of b: Now, we need to find the value of $b$ in the expression $(x + b)^2$ that will complete the square. The term $b$ is half of the coefficient of $x$ , which is $-2$ in this case. $\newline$ So, $b = -2 / 2 = -1$ .
Add/Subtract $b^2$ : We add and subtract $b^2$ inside the parentheses to complete the square. Since $b = -1$ , $b^2 = (-1)^2 = 1$ . We add and subtract $1$ inside the parentheses, which gives us $-2(x^2 - 2x + 1 - 1) + 6$ .
Write as Perfect Square: Now we can write the expression as a perfect square: $-2((x - 1)^2 - 1) + 6$ .
Distribute $-2$ : Next, we distribute the $-2$ inside the parentheses: $-2(x - 1)^2 + 2 + 6$ .
Combine Constant Terms: Finally, we combine the constant terms: $-2(x - 1)^2 + 8$ .

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